Bouncing ball orbits and symmetry breaking effects in a three-dimensional chaotic billiard

TL;DR

This paper analyzes the classical and quantum properties of a 3D chaotic billiard with symmetry breaking, revealing how geometric differences influence spectral statistics and the emergence of stable orbits.

Contribution

It introduces an analytic method for nongeneric periodic orbits and explores symmetry breaking effects on spectral properties in a 3D stadium billiard.

Findings

Level statistics match random matrix theory for unequal radii.

Symmetry breaking causes deviations in spectral statistics.

Stable and marginally stable orbits appear when radii are equal.

Abstract

We study the classical and quantum mechanics of a three-dimensional stadium billiard. It consists of two quarter cylinders that are rotated with respect to each other by 90 degrees, and it is classically chaotic. The billiard exhibits only a few families of nongeneric periodic orbits. We introduce an analytic method for their treatment. The length spectrum can be understood in terms of the nongeneric and unstable periodic orbits. For unequal radii of the quarter cylinders the level statistics agree well with predictions from random matrix theory. For equal radii the billiard exhibits an additional symmetry. We investigated the effects of symmetry breaking on spectral properties. Moreover, for equal radii, we observe a small deviation of the level statistics from random matrix theory. This led to the discovery of stable and marginally stable orbits, which are absent for un equal radii.