Scattering matrices and scattering geodesics of locally symmetric spaces

Abstract

Let Γ/ X be a Q -rank one locally symmetric space. We describe the frequencies of oscillation of scattering matrices on Γ/ X in the energy variable in terms of sojourn times of scattering geodesics. Scattering geodesics are the geodesics which move to infinity in both directions and are distance minimizing near both infinities. The sojourn time of a scattering geodesic is the time it spends in a fixed compact region. The frequencies of oscillation come from the singularities of the Fourier transforms of scattering matrices and we show that these occur at sojourn times of scattering geodesics on the locally symmetric space. This generalizes a result of Guillemin obtained in the case of finite volume noncompact Riemann surfaces.