Classical and quantum mechanics of a chaotic hamiltonian system: the wedge billiard

We examine the correspondence between a quantum system and its classical analogue. We consider the wedge billiard which shows a wide variety of behaviors ranging from integrable to strongly chaotic. Angle-action coordinates are provided for the flow and for the bouncing map when the dynamics is integrable. Special sets of orbits, such as orbits bouncing in the corner and orbits sliding along the boundary are investigated. Their influence on the classical statistical properties is also discussed. At the quantum level, we extend the scattering approach to quantize general Hamiltonian systems in 2 dimensions and demonstrate it for the wedge. The resulting energy levels are given by the zeros of a secular equation. The corresponding wave functions may also be constructed. We verify analytically that this quantization condition is exact in the case of the integrable wedges. This method has been applied successfully to compute numerically a few thousand levels. Using this data, we can check the Gutzwiller trace formula and other semiclassical relations involving the scattering matrix and periodic orbits. Finally we perform a detailed statistical analysis of several velocity and curvature distributions.

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