Classical Quantization of a Hamiltonian with Ergodic Behavior

Conservative Hamiltonian systems with two degrees of freedom are discussed where a typical trajectory fills the whole surface of constant energy. The trace of the quantum mechanical Green's function is approximated by a sum over classical periodic orbits. This leads directly to Selberg's trace formula for the motion of a particle on a surface of constant negative curvature, and, when applied to the anisotropic Kepler problem, yields excellent results for all the energy levels.