Generic spectral properties of right triangle billiards

Abstract

This article presents a new method to calculate eigenvalues of right triangle billiards. Its efficiency is comparable to the boundary integral method and more recently developed variants. Its simplicity and explicitness however allow new insight into the statistical properties of the spectra. We analyse numerically the correlations in level sequences at high level numbers (>10^5) for several examples of right triangle billiards. We find that the strength of the correlations is closely related to the genus of the invariant surface of the classical billiard flow. Surprisingly, the genus plays and important role on the quantum level also. Based on this observation a mechanism is discussed, which may explain the particular quantum-classical correspondence in right triangle billiards. Though this class of systems is rather small, it contains examples for integrable, pseudo integrable, and non integrable (ergodic, mixing) dynamics, so that the results might be relevant in a more general context.