Relationship between the Discrete and Resonance Spectrum for the Laplace Operator on a Noncompact Hyperbolic Riemann Surface

Abstract

We consider arbitrary noncompact hyperbolic Riemann surfaces of finite area. For such surfaces, we obtain identities relating the discrete spectrum of the Laplace operator to the resonance spectrum (formed by the poles of the scattering matrix). These identities depend on the choice of a test function. We indicate a class of admissible test functions and consider two examples corresponding to specific choices of the test function.