A trace formula for nodal counts: Surfaces of revolution
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A trace formula for nodal counts:Surfaces of revolution
Sven GnutzmannPanos KarageorgiU. S.
Rehovot, April 2006
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Reminder: The spectral trace formulaor
how to count the spectrum
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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
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The sequence of nodal counts
Sturm (1836) : For d=1 : n = n Courant (1923) : For d>1 : n n
n=8
n =20
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Counting Nodal Domains: Separable systemsRectangle, Disc “billiards” in R2
Surfaces of revolutionLiouville surfaces
Main Feature – Checkerboard structure
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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
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Bohr Sommerfeld (EBK) quantization
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Nodal counting
Order the spectrum using the spectral counting function:
The nodal count sequence :
The cumulative nodal count:
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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)
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A trace formula for the nodal sequence
Cumulative nodal counting
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k
k
Numerical simulation: the smooth termEllipsoid of revolution
c(k)~a k2 )c(k) – a k2/(k2
k
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The fluctuating part = c(k) - smooth (k)
Correct power-law
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The scaled fluctuating part:
Its Fourier transform =the spectrum of periodic orbits lengths
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The main steps in the derivation
Poisson summation
Semi-classical (EBK)
n+1/2 ! n
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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
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The oscillatory term
Saddle point integration:
Picks up periodic tori with action:
Collecting the terms one gets the trace formula
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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
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Nodal domains are created or merged by fission or fusion