Point Perturbations of Integrable Billiards Physics@Technion点扰动的积台球物理”技术的研究.ppt

Point Perturbations of Integrable Billiards ******@Technion点扰动的积台球物理”技术的研究
Outline
Motivation: spectral statisticsring theory:
(without the boundary)
Geometrical theory of diffraction, Keller.
“Mathematical” point of view:
The self-adjoint extension of a
Hamiltonian
One can define a family of extensions, with a simple Green function:
Zorbas
is related to the scattering strength
The new eigenvalues are the poles of
For closed systems:
A quantization condition for new eigenvalues
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
In the limit of infinite number of bonds with random bond lengths
The spectral statistics of star graphs are those of Seba billiard with
Periodic orbit calculation of spectral statistics
Reminder:
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
Diffracting orbits with n segments
Sieber
Can one find diffracting orbit with the same length of a periodic orbit?
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
Results:
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
Dependence on location:
For the rectangular billiard the spectral statistics depend in a complicated manner on the location of the perturbation:
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 function