Energy-level statistics of an integrable billiard system in a rectangle in the hyperbolic plane

Abstract

A separable billiard system in the hyperbolic strip, a particular realization of the hyperbolic plane, is investigated. The author considers a rectangle bounded by straight lines in the hyperbolic strip as an approximation of either a regular octagon with area A=4 pi , or of a hyperbolic rectangle bounded by geodesics, respectively. Due to the simple geometry of the approximation the Schrodinger equation is separable, and leads to simple transcendental equations for the eigenvalues in terms of Legendre functions. The statistical properties of the eigenvalue spectrum are investigated and checked with respect to Weyl's (1912) law, level spacing, number variance and spectral rigidity, respectively. In particular the numerical results for the spectral rigidity are in close agreement with the semi-classical theory of Berry (1985).