Energy Fluctuation Analysis in Integrable Billiards in Hyperbolic Geometry

Abstract

In this contribution I provide new examples of integrable billiard systems in hyperbolic geometry. In particular, I present one billiard system in the hyperbolic plane, called “Circular billiard in the Poincare disc”, and one three-dimensional billiard, called “Spherical billiard in the Poincare ball”. In each of the billiard systems, the quantization condition leads to transcendental equations for the energy eigen-values E n , which must be solved numerically. The energy eigen-values are statistically analyzed with respect to spectral rigidity and the normalized fluctuations about Weyl’s law. For comparison, some flat two- and three-dimensional billiard systems are also mentioned. The results found are in accordance with the semiclassical theory of the spectral rigidity of Berry, and the conjecture of Steiner et al. concerning the normalized fluctuations for integrable billiard systems.