Resonance behaviour for classes of billiards on the Poincaré half-plane

Author: Phil Howard

Reference: RHUL-MA-2007-6

Abstract:

The classical and quantum mechanics of two linked classes of open billiard systems on the Poincaré half-plane is studied. These billiard systems are presented as models of arithmetic scattering systems under deformation. An investigation is made of the classical phase space and the stability of a certain family of periodic orbits is investigated. The movement of the positions and widths of the resonances is followed, as the shape of Artinís billiard is deformed. One deformation varies its lower boundary, interpolating between integrable and fully chaotic cases. The other deformation translates the right hand wall of the billiard, thus interpolating between several examples of Hecke triangle billiards. The variation of the statistical properties of the spectra are focussed on and the transitions in the statistics of the resonance positions and widths are mapped out in detail near particular values of the deformation parameters, where the billiard is a fundamental domain for some arithmetic group. Analytic solutions for the scattering matrix and the resonance positions in these particular systems are derived and numerical results are obtained which are in excellent agreement with the predictions.

Away from the arithmetic systems, both generic behaviour according to the predictions of Random Matrix Theory, and non-generic behaviour is found, with deviations occurring particularly in the long range statistics. In the integrable case, semiclassical WKB theory is used to produce accurate wavefunctions and eigenvalues. For the general deformation a number of numerical methods are explored, such as the finite element method, complex absorbing potentials and collocation, in order to find an optimum method to locate the resonance positions.