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Localized Perturbations of Localized Perturbations of Integrable BilliardsIntegrable Billiards
Saar RahavSaar Rahav
Technion, Haifa, May 2004Technion, Haifa, May 2004
OutlineOutline
• Motivation: spectral statistics and universality
• What is a perturbation at a point• Two ways for spectral statistics• Connection to star graphs• Periodic orbits and spectral statistics• Dependence on location of perturbation• Conclusions
Spectral statistics of dynamical systems
Dynamical systems exhibit spectral statistics of “random matrices”
• Time reversal symmetry statistics of random real symmetric matrices (GOE)
• No time reversal symmetry random Hermitian matrices (GUE)
•Integrable systems Poissonian statistics (uncorrelated levels)
Spectral statistics do not depend on details: Universality
How to explain this universality?
Some properties of the dynamical system itself must be universal (sum rules).
Examples for different dynamical systems:
Sinai billiard - chaotic
Ray splitting – pseudointegrable, non universal
Šeba billiard – singular billiard
Non universal
The statistical measures
Nearest neighbor spacing distribution ( )P s
Spectral form factor ( , ) ( ) ( ) i tosc oscK E t d d E d E e
Semiclassical analysis of spectral statistics
Trace formulae:
( ) exp ( ) /osc p pp
d E d A iS E Gutzwiller,
Berry and Tabor
Relates the density of states and periodic orbits
For large energies (semiclassical limit) pS
Contributions with random phases in the cosine
The sum is dominated by almost equal actions:p pS S
The semiclassical Form factor
*
,
( ) cos ( ) / 2p pp p p p
p p
S SK t A A t T T
Berry
The diagonal approximation:
2( ) j jj
K t A t T
Take only pairs with the same action i jS S
Berry
Evaluated using the Hannay & Ozorio de Almeida sum rule
GUE
( ) 2 GOE
1 Integrable
K
The correct short time asymptotics !
Validity of RMT Sum rules for periodic orbitsHigher order terms? corrections GOE were calculated by Sieber and Richter, see also Muller, Haake, et. al.
What are singular billiards?
“Physical” point of view:
Integrable Quantum systems, with local perturbation
a L
0r
The diffraction constant ( , ) ( )D E k T q
is proportional to the scattering amplitude and satisfy the optical theorem
2
2
0
1, ,
8D d D
The perturbation can be described be means of scattering theory:
0 00 0 0( , , ) ( , , ) ( , , ) ( , ) ( , , )G r r k G r r k G r r rk D G r k (without the boundary)
Geometrical theory of diffraction, Keller.
“Mathematical” point of view:
The self-adjoint extension of a Hamiltonian 0H r
One can define a family of extensions, with a simple Green function:
0 0( , ; ) , ; ( , ) ( , , ) ( , , )G r r z G r r z T z G r r z G r r z Zorbas
1
0 0 0 0( , ) 1 ( ) ( , , ) ( , , ) ( ) ( , , ) ( , , )i iT z e i z dr G r r z G r r i e i z dr G r r z G r r i
is related to the scattering strength
The new eigenvalues are the poles of ,T z
For closed systems: 1
( ) ( ), , n n
n n
r rG r r z
z E
A quantization condition for new eigenvalues
2 20 02 2
sin 1 1( ) ( ) 0
1 cos 1 1n
n nn nn n n
Er r
E z E E
Why singular billiards?
•Dynamics intermediate between integrable and chaotic
•Important diffraction effects
•Simple system
•What is the spectral statistics?
•New universality class for spectral statistics?
•A new ‘test’ for periodic orbit theory
Two approaches for spectral statistics of singular billiards
1. Periodic orbits
Simple scattering without a boundary
0 0 0 00( , , ) ( , , ) ( , , ) ( , , )G r r k G r r k G r rr k DG r k
The boundary can be added using an integral equation
1 1ˆ ˆ ˆ1 1 1 2 1
1
1 12 ( , ) ( , ) ( , )
n n
n
osc n n n n n n nn
dd ds ds G r r G r r G r r
n dk
In the semiclassical limit,
0
1( , , ) ik r rG r r k e
k r r
The integrals over the boundary are dominated by contributions that perform specular reflections
Rahav, Fishman
Bogomolny, Giraud
The integrals lead to two types of orbits:
Periodic orbits - do not hit the scatterer
Orbits with segments which start and end at r0 – diffracting orbits
jl
leads to a modified trace formula:
1 2
1 2
1 2
( )(1) (2),
,
( ) . .j jp j ik l likl iklosc p j j j
p j j j
d E A e A e A e c c
With (1)A D (2) 2A D
More diffractions: more powers of 1/ kl Higher powers of more segments Non diagonal contributions
2. Ensemble averaging of the quantization condition
20
1
( ) 1N
n n
r
z E
Approximately:
Bogomolny, Gerland, Giraud, Schmit
Properties: •LHS has poles at ‘unperturbed’ energy levels
•LHS monotonically decreasing with z
•Exactly one solution in 1,i iE E
The density of states is:2 2 2 2
0 0 0 02 2
1 1 1 1
( ) ( ) ( ) ( )( ) 1 exp 1
( ) 2 ( )
N N N Nn n n n
n k n kn n n n
r r r rdd E i
E E E E E E E E
Integrable system nE are independent random variables
The distribution of 2
0( )n r is uncorrelated with nE
One can build statistical measures, e.g. ( ) ( )d E d E And average over the unperturbed energies and wavefunction values
A kind of ensemble averageAdvantage – the integrals separate into independent farctors
Results: (simplified)
4
rectangle, periodic BC( )
ln Dirichlet BC
SP S
S S
Level repulsion 0S
1( ) CSP S e
S S Exponential falloff
Intermediate statistics
Connection to star graphs
Quantum graphs: Kottos, Smilansky
Free motion on bonds, boundary conditions on vertices
Star Graphs: Berkolaiko, Bogomolny, Keating
For star graphs, the quantization condition is
1
1tan
N
jj
l k
In the limit of infinite number of bonds with random bond lengths
The spectral statistics of star graphs are those of Seba billiard with 2
0( ) Constn r
Periodic orbit calculation of spectral statistics
Reminder: *
,
( ) cos ( ) / 2p pp p p p
p p
S SK t A A t T T
Where the lengths may be composed of several diffracting segments
What types of contributions may survive?
For the rectangular billiard:
Diagonal contributions:
The periodic orbits contribute ( ) 1K
Diffracting orbits with n segments2 1( )n nK D Sieber
Can one find diffracting orbit with the same length of a periodic orbit?
Yes. A forward diffracting orbit!
A ‘kind’ of diagonal contribution:21
( )4
K D D
Non diagonal contributions:
21 1 1
2k l kt
x l x
,l x kForThe difference in phase is small for 1t
There are many (~k) such contributions
Results:
Scatterer at the center2
4 4 62 3 41 1 1( ) 1 ( )
4 8 2 24
DK D D D O
Typical location of scatterer: 2
4 4 62 3 49 81 25( ) 1
4 128 512 1536
DK D D D O
All form factors start at 1 and exhibit a dip before going back to 1.
Intermediate statistics
Dependence on location:
For the rectangular billiard the spectral statistics depend in a complicated manner on the location of the perturbation:
Complementary explanations:
1. Degeneracies in lengths of diffracting orbits
2. The distribution of values of wavefunctions:
0 0 0( ) sin sinn r mx nya b
Differs if are rational or not0 0,x y
a b
Is such behavior typical?
The Circle billiard:
Angular momentum conservation min2
Lr r
mE
Quantum wave functions are exponentially small for minr r
20
1
( ) 1N
n n
r
z E
So for exponentially small wavefunction the
eigenvalues are almost unchanged
The spectrum of the singular circle billiard can be (approximately) divided into two components:
1. Almost unperturbed spectrum, composed of wavefunctions localized on r>r0.
2. Strongly perturbed spectrum.
How many levels are unperturbed?
1 0 0 0unpertubed
2cos 1
r r rX
R R R
Superposing the two spectra:
The statistics depend on the location of the scatterer.
Partial level repulsion?
ConclusionsConclusions
• The spectral statistics differ from known universality classes – Intermediate statistics
• Strong contribution due to diffraction – non classical
• Statistics depend on location of perturbation – non universal
• However, the statistics of different singular billiards show similarities
• The wavefunctions are not ergodic (Berkolaiko, Keating, Marklof, Winn)
Interesting open problemsInteresting open problems
• Understanding pseudointegrable systems, where the diffraction contributions are non uniform
• Resummation of the series for the form factor• Understanding singularities of form factors• Better understanding of wavefunctions• Dependence on number of scatterers