Scattering matrices and scattering geodesics of locally symmetric spaces

Let Gamma /X be a Q-rank one locally symmetric space. We describe the frequ encies of oscillation of scattering matrices on Gamma /X in the energy vari able in terms of sojourn times of scattering geodesics. Scattering geodesic s are the geodesics which move to infinity in both directions and are dista nce minimizing near both infinities. The sojourn time of a scattering geode sic is the time it spends in a fixed compact region. The frequencies of osc illation come from the singularities of the Fourier transforms of scatterin g matrices and we show that these occur at sojourn times of scattering geod esics on the locally symmetric space. This generalizes a result of Guillemi n obtained in the case of finite volume noncompact Riemann surfaces. (C) 20 01 Editions scientifiques et medicales Elsevier SAS.