Classical and quantum billiards : integrable, nonintegrable and pseudo-integrable

Statistical properties of the spectra of quantum two dimensional billiards are shown to be linked to the nature of the dynamics of the corresponding classical systems. Quantised pseudo-integrable billiard exhibits level repulsion, in spite of non chaotic dynamics of its classical counterpart. We conjecture that the level statistics of a quantum pseudo-integrable system depends on the genus of the invariant manifold equivalent to its classical phase space. A model of billiards with finite walls suitable to investigate the problems of chaotic scattering is proposed.