patterns in spectra - mathphysics.comharrell-stubbe tams 1996. statistics of spectra a reverse...
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Universal patterns in spectra of differential operators
Copyright 2008 by Evans M. Harrell II.
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This lecture will not involve snakes, machetes, or other dangerous objects.
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Laplace, Laplace-Beltrami, and Schrödinger
H = T + V(x)
-
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Laplace, Laplace-Beltrami, and Schrödinger
H uk = λk uk
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Musicians’ vibrating membranes
Inuit
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Musicians’ vibrating membranes
Inuit
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Musicians’ vibrating membranes
Inuit
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Musicians’ vibrating membranes
Inuit
West Africa
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Musicians’ vibrating membranes
Inuit
West Africa
USA
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Physicists’ vibrating membranes
Sapoval 1989
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Physicists’ vibrating membranes
Sapoval 1989
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Physicists’ vibrating membranes
Sapoval 1989
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Mathematicians’ vibrating membranes
Wolfram MathWorld, following Gordon, Webb, Wolpert. (“Bilby and Hawk”)
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Mathematicians’ vibrating membranes
Wolfram MathWorld, following Gordon, Webb, Wolpert. (“Bilby and Hawk”)
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Quantum waveguides - same math (sort of), different physics
Ljubljana Max Planck Inst.
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There’s something Strang about this column.
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Should you believe this man? For the 128-gon,
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Should you believe this man? For the 128-gon,
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Should you believe this man? For the 128-gon,
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But according to the Ashbaugh-Benguria Theorem, λ2/ λ1 ≤ 2.5387…!
(In 2 D)
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Aha!
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Universal spectral patterns
The top of the spectrum is controlled by the bottom!
Example: Two classic results relate the spectrum to the volume of the region. For definiteness, consider
H= -∇2 , Dirichlet BC (i.e., functions vanish on ∂Ω).
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A simple universal spectral pattern (for the Dirichlet Laplacian)
The Weyl law (B=std ball): λk (Ω) ∼ Cd (|B|k /|Ω|)2/d as k→∞. Faber-Krahn: λ1(Ω) ≥ λ1(B) (|B|/|Ω|)2/d . Therefore, universally, lim (λk/λ1 k2/d) ≤ Cd/λ1(B). Equality only for the ball.
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Statistics of spectra
For Laplacian on a bounded domain,
Harrell-Stubbe TAMS 1996
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Statistics of spectra
A reverse Cauchy inequality:
The variance is dominated by the square of the mean.
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Statistics of spectra
Dk :=
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Means of ratios:
Harrell-Hermi preprint 2007
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Reverse arithmetric-geometric mean inequalities:
For Laplacian on a bounded domain, 0 < r ≤ 3,
Harrell-Stubbe in prep.
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Surfaces and other “immersed manifolds”
El Soufi-Harrell-Ilias preprint 2007
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Surfaces and other “immersed manifolds”
Harrell CPDE 2007
For a Schrödinger operator T+V(x) on a closed hypersurface,
λ2 – λ1 ≤
El Soufi-Harrell-Ilias preprint 2007
Statistical bounds for eigenvalue gaps, which are all sharp in the case V = g h2, standard sphere.
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Surfaces and other “immersed manifolds”
El Soufi-Harrell-Ilias preprint 2007
For Laplace-Beltrami:
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Sum rules and Yang-type inequalities
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“Universal” constraints on eigenvalues
H. Weyl, 1910, Laplace operator λn ~ n2/d W. Kuhn, F. Reiche, W. Thomas, 1925, sum
rules for energy eigenvalues of Schrödinger operators
L. Payne, G. Pólya, H. Weinberger, 1956: gaps between eigenvalues of Laplacian controlled by sums of lower eigenvalues:
E. Lieb and W. Thirring, 1977, P. Li, S.T. Yau, 1983, sums of eigenvalues, powers of eigenvalues.
Hile-Protter, 1980, stronger but more complicated analogue of PPW
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“Universal” constraints on eigenvalues
Ashbaugh-Benguria 1991, isoperimetric conjecture of PPW proved.
H. Yang 1991-5, unpublished, complicated formulae like PPW, respecting Weyl asymptotics.
Harrell, Harrell-Michel, Harrell-Stubbe, 1993-present, commutators.
Hermi PhD thesis, articles by Levitin and Parnovsky.
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PPW – the grandfather of many universal spectral patterns:
All eigenvalues are controlled by the ones lower down! (Payne-Pólya-Weinberger, 1956)
Not so far off Ashbaugh-Benguria (In 2D, λ2/λ2 ≤ 3 rather than 2.5387….
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Universal spectral patterns
Numerous extensions by same method: Cook up a trial function for min-max by multiplying u1, … uk by a coordinate function xα. For example:
Hile-Protter 1980 Compare to PPW:
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Universal spectral patterns
Numerous extensions by same method: Cook up a trial function for min-max by multiplying u1, … uk by a coordinate function xα.
Before 1993, these were all lousy for large k, as we knew by Weyl.
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H.C. Yang, unpublished preprint, 1991,3,5
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H.C. Yang, unpublished preprint, 1991,3,5
Some people will do anything for an eigenvalue!
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H.C. Yang, unpublished preprint, 1991,3,5
If you estimate with the Weyl law λj (Ω) ∼ j2/d , the two sides agree with the correct constant.
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What is the underlying idea behind these universal spectral relations and how can you
extract consequences you can understand?
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Commutators of operators
[H, G] := HG - GH [H, G] uk = (H - λk) G uk If H=H*, <uj,[H, G] uk> = (λj - λk) <uj,Guk>
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Commutators of operators
[G, [H, G]] = 2 GHG - G2H - HG2 Etc., etc. Try this one:
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Commutators of operators
For (flat) Schrödinger operators, G=xα, [H, G] = - 2 ∂/∂xα [G, [H, G]] = 2
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Commutators of operators
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Commutators of operators
Compares with Hile-Protter:
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Commutators of operators
Variant: If H is the Laplace-Beltrami operator + V(x), there is an additional term involving mean curvature.
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What is the underlying idea behind these universal spectral relations, and how can you
extract consequences you can understand?
The counting function, N(z) := #(λk ≤ z)
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What is the underlying idea behind these universal spectral relations, and how can you
extract consequences you can understand?
The counting function, N(z) := #(λk ≤ z) Integrals of the counting function,
known as Riesz means (Safarov, Laptev, etc.):
Chandrasekharan and Minakshisundaram, 1952
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Trace identities imply differential inequalities
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Trace identities imply differential inequalities
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Trace identities imply differential inequalities
It follows that R2(z)/z2+d/2 is an increasing function.
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Understandable eigenvalue bounds from inequalities on Rρ(z):
Legendre transform – gives the ratio bounds
Laplace transform – gives a lower bound on Z(t) = Tr(exp(-Ht)).
The latter implies a lower bound on the spectral zeta function (through the Mellin transform)
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Summary
Eigenvalues fall into universal patterns, with distinct statistical signatures
Spectra contolled by “kinetic energy” and geometry
Underlying idea: Identities for commutators
Extract information about λk by differential inequalities, transforms.