Quantising a generic family of billiards with analytic boundaries

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Abstract
A generic family of plane billiards has been discovered recently. The shape of the boundary is given by the quadratic conformal image of the unit circle, and is thus real analytic. For small deformations of the unit disc the billiard is a typical KAM system, but becomes ergodic or even mixing when the curvature of the boundary vanishes at some point. The Kolmogorov entropy has been calculated, and it increases with the deformation of the boundary. The author studies aspects of the quantum chaos for this billiard. He solves numerically the eigenvalue problem for the Laplace operator with Dirichlet's boundary condition. He examines the spectrum, and inspects the avoided crossings at which mixing of nearby states occurs. The variation of the nodal structure and of the localisation properties of the eigenfunctions is studied. In analysing the level spacing distribution he finds a continuous transition from the Poisson distribution towards the Wigner distribution. The exponent in the level repulsion law varies continuously along with a generic perturbation. For small perturbations it seems to be proportional to the square root of the perturbation parameter.

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