A trace formula for nodal counts:Surfaces of revolution

Sven GnutzmannPanos KarageorgiU. S.

Rehovot, April 2006

Reminder: The spectral trace formulaor

how to count the spectrum

The spectral counting function:

Trace formula :

Oscillatory

A periodic orbit

1 2 3 4 5 6E

1

2

3

4NE

The geometrical contents of the spectrum

Smooth

The sequence of nodal counts

Sturm (1836) : For d=1 : n = n Courant (1923) : For d>1 : n n

n=8

n =20

Counting Nodal Domains: Separable systemsRectangle, Disc “billiards” in R2

Surfaces of revolutionLiouville surfaces

Main Feature – Checkerboard structure

Simple Surfaces of Revolution (SSR)

0 0.25 0.5 0.75 1m

1

n

0 0.25 0.5 0.75 1

1

The curve Hn,m1 in the action variable plane . Green ,1ו: Blue 0.5ו: , Orange 0.1ו: , Light Blue 1.8ו:

n(m)

m

for a few ellipsoids

simple surfaces: n’’(m) 0

Bohr Sommerfeld (EBK) quantization

Nodal counting

Order the spectrum using the spectral counting function:

The nodal count sequence :

The cumulative nodal count:

1 2 3 4E

1

2

3

4

5

6NE

1 2 3 4 5 6 7k

2

4

6

8

10

12

14

CkC(k)

Cmod(k)

A trace formula for the nodal sequence

Cumulative nodal counting

k

k

Numerical simulation: the smooth termEllipsoid of revolution

c(k)~a k2 )c(k) – a k2/(k2

k

The fluctuating part = c(k) - smooth (k)

Correct power-law

The scaled fluctuating part:

Its Fourier transform =the spectrum of periodic orbits lengths

The main steps in the derivation

Poisson summation

Semi-classical (EBK)

n+1/2 ! n

Change of variables:

Approximate:

Integration limit:

0 0.25 0.5 0.75 1m

1

n

0 0.25 0.5 0.75 1

1

The curve Hn,m1 in the action variable plane . Green ,1ו: Blue 0.5ו: , Orange 0.1ו: , Light Blue 1.8ו:

Another change of variables

The oscillatory term

Saddle point integration:

Picks up periodic tori with action:

Collecting the terms one gets the trace formula

Closing remarks :What is the secret behind nodal counts for separable systems?

Consider the rectangular billiard:

E(n,m)= n2 + m2 ; (n,m)= n m

~ (Lx / Ly)2

Follow the nodal sequence as a function of :

At every rational value of there will be

pairs of integers (n1,m1) and (n2,m2) for which the eigen-values cross:

- +E (n1,m1) < E (n2,m2) ; E (n1,m1) = E (n2,m2) ; E (n1,m1) > E (n2,m2)

! at this the nodal sequence will be swapped !

Thus: The swaps in the nodal sequence reflect the

the value of ! Geometry of the boundary

Nodal domains are created or merged by fission or fusion