Chaotic Scattering on Noncompact Surfaces of Constant Negative Curvature

Abstract

In this paper we consider the problem of quantizing the geodesic motion on noncompact surfaces of constant negative curvature. This problem can be regarded as a model of multichannel quantum scattering. Knowing that the geodesic motion on such surfaces is chaotic, we examine how the chaos of the underlying classical dynamics manifests itself in the corresponding quantum system. We calculate the scattering matrix, and introduce the associated time delays. With the help of Selberg's trace formula we establish a connection between the classical periodic orbits and the quantum resonances and energy eigenvalues. Illustrative examples for a class of $\sum_{g,2}$ surfaces are given