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Localized Perturbations of Integrable Billiards
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  1. Localized Perturbations of Integrable Billiards Saar Rahav Technion, Haifa, May 2004

  2. Outline • 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

  3. Spectral statistics do not depend on details: Universality How to explain this universality? 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) Some properties of the dynamical system itself must be universal (sum rules).

  4. Sinai billiard - chaotic Ray splitting – pseudointegrable, non universal Šeba billiard – singular billiard Non universal Examples for different dynamical systems:

  5. Nearest neighbor spacing distribution Spectral form factor The statistical measures

  6. Trace formulae: Gutzwiller, Berry and Tabor Relates the density of states and periodic orbits The semiclassical Form factor Berry For large energies (semiclassical limit) Contributions with random phases in the cosine The sum is dominated by almost equal actions: Semiclassical analysis of spectral statistics

  7. Take only pairs with the same action Evaluated using the Hannay & Ozorio de Almeida sum rule The correct short time asymptotics ! Higher order terms? corrections GOE were calculated by Sieber and Richter, see also Muller, Haake, et. al. Validity of RMT Sum rules for periodic orbits The diagonal approximation: Berry

  8. The perturbation can be described be means of scattering theory: (without the boundary) Geometrical theory of diffraction, Keller. The diffraction constant is proportional to the scattering amplitude and satisfy the optical theorem What are singular billiards? “Physical” point of view: Integrable Quantum systems, with local perturbation

  9. The self-adjoint extension of a Hamiltonian One can define a family of extensions, with a simpleGreen function: Zorbas is related to the scattering strength The new eigenvalues are the poles of For closed systems: A quantization condition for new eigenvalues “Mathematical” point of view:

  10. 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

  11. The boundary can be added using an integral equation In the semiclassical limit, The integrals over the boundary are dominated by contributions that perform specular reflections Two approaches for spectral statistics of singular billiards Rahav, Fishman Bogomolny, Giraud 1. Periodic orbits Simple scattering without a boundary

  12. leads to a modified trace formula: With Higher powers of More diffractions: more powers of more segments Non diagonal contributions 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

  13. Properties: • LHS has poles at ‘unperturbed’ energy levels • LHS monotonically decreasing with z • Exactly one solution in 2. Ensemble averaging of the quantization condition Bogomolny, Gerland, Giraud, Schmit Approximately:

  14. are independent random variables Integrable system is uncorrelated with The distribution of One can build statistical measures, e.g. And average over the unperturbed energies and wavefunction values Advantage – the integrals separate into independent farctors A kind of ensemble average Results: (simplified) Level repulsion Exponential falloff Intermediate statistics The density of states is:

  15. Star Graphs: Berkolaiko, Bogomolny, Keating For star graphs, the quantization condition is In the limit of infinite number of bonds with random bond lengths The spectral statistics of star graphs are those of Seba billiard with Connection to star graphs Quantum graphs: Kottos, Smilansky Free motion on bonds, boundary conditions on vertices

  16. Diagonal contributions: The periodic orbits contribute Diffracting orbits with n segments Sieber Periodic orbit calculation of spectral statistics Reminder: Where the lengths may be composed of several diffracting segments For the rectangular billiard: What types of contributions may survive? Can one find diffracting orbit with the same length of a periodic orbit?

  17. Yes. A forward diffracting orbit! A ‘kind’ of diagonal contribution: Non diagonal contributions: For The difference in phase is small for There are many (~k) such contributions

  18. Scatterer at the center Typical location of scatterer: All form factors start at 1 and exhibit a dip before going back to 1. Intermediate statistics Results:

  19. Complementary explanations: • Degeneracies in lengths of diffracting orbits • The distribution of values of wavefunctions: Differs if are rational or not Is such behavior typical? The Circle billiard: Angular momentum conservation Quantum wave functions are exponentially small for Dependence on location: For the rectangular billiard the spectral statistics depend in a complicated manner on the location of the perturbation:

  20. The spectrum of the singular circle billiard can be (approximately) divided into two components: • Almost unperturbed spectrum, composed of wavefunctions localized on r>r0. • Strongly perturbed spectrum. How many levels are unperturbed? Superposing the two spectra: The statistics depend on the location of the scatterer. Partial level repulsion? So for exponentially small wavefunction the eigenvalues are almost unchanged

  21. Conclusions • 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)

  22. Interesting 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