Quantum billiards in reduced phase space

Abstract

This work is embedded in the context of quantum chaos. We show in this work that the results known for two-dimensional billiard systems can also be proven for a system that is reduced to one dimension. In both the classical and the quantum mechanical system the billiard boundary acts as a global Poincaré section. In this context, the boundary integral operator and the normal derivatives of the solutions of the Helmholtz equation play a central role. For the derivations in reduced phase space we first modify the formalism of coherent states and of the anti-Wick quantization such that both can be used on the billiard boundary. With the help of these methods we can calculate the unitary part of the boundary integral operator and show that the semiclassically leading part of this operator coincides with the Bogomolny operator. Furthermore, we find a semiclassical relation between the boundary functions and the eigenfunctions of the Bogomolny operator. Additionally, we prove a quantum ergodicity theorem on the billiard boundary and examine the mean behaviour of the boundary functions in the semiclassical limit.