Quantum chaos and the Selberg trace formula on Riemann surfaces

The classical dynamics of a point particle sliding freely on a Riemann surface (RS) of constant negative curvature (hyperbolic flow) is strongly chaotic (ergodic, mixing and Bernoullian). The corresponding quantum dynamics is given by the eigenvalue problem of the Laplace-Beltrami operator on the given RS. The Selberg trace formula is a deep relation in spectral geometry which expresses the quantal energy spectrum by the length spectrum of the classical periodic orbits (closed geodesics) on the given RS. For compact RSs of genus two (arithmetic and non-arithmetic ones) and the non-compact modular surface, we present analytical and numerical results on the length spectrum, the eigenvalues and eigenfunctions, spectral statistics and a comparison with random matrix theory. Finally, we discuss a conjecture on the value distribution of the Selberg zeta function on the critical line.