spectral functions, the geometric power of eigenvalues,

Report

Post on 04-Jul-2015

369 Views

Category:

Education

4 Downloads

Preview:

Click to see full reader

TRANSCRIPT

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral FunctionsThe Geometric Power of Eigenvalues

Pedro Fernando Morales

Department of MathematicsBaylor University

pedro moralesbayloredu

Athens Ohio 10182012

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ

(eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ

(eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ

(eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ

(eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ

(eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ

(eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ

(eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ

(eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ

(eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ

(eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ

(eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ

(eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ

(eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ

(eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ

(eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ

(eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ

(eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ

(eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ

(eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

top related

spectral coarsening of geometric...
Documents
sgt toolbox a playground for spectral graph theory ·...
Documents
eigenvalues and geometric representations of graphs lászló...
Documents
an experimental study into spectral and geometric...
Documents
selected titles in this series · spectral variables for...
Documents
the spectral radius of connected graphs with the ... · the...
Documents
lectures on spectral graph theoryfan/cbms.pdfchapter 1...
Documents
eigenvalues and geometric representations of graphs l á...
Documents
scientific computing seminar amls, spectral schur...
Documents
a compact spectral descriptor for shape...
Documents
spectral methods for mesh processing and...
Documents
some geometric bounds on eigenvalues of elliptic pdes evans...
Documents
spectral properties of self-adjoint matrices contents 5101...
Documents
geometric aspects of optimization and applications to...
Documents
a geometric proof of the spectral theorem for unbounded self
Documents
geometric analysis and spectral...
Documents
a geometric approach to spectral subtraction ·...
Documents
constrained clustering by spectral kernel...
Documents
a geometric approach to spectral subtraction," speech
Documents
spectral graph theory and clustering - university of...
Documents