SEMICLASSICAL WAVE PROPAGATION FOR LARGE TIMES

Abstract. We study solutions of the time dependent Schrödinger equation on Riemann-ian manifolds with oscillatory initial conditions given by Lagrangian states. Semiclassical approximations describe these solutions for ~ → 0, but their accuracy for t → ∞ is in general only understood up to the Ehrenfest time T ∼ ln 1/~, and the most difficult case is the one where the underlying classical system is chaotic. We show that on surfaces of constant negative curvature semiclassical approximations remain accurate for times at least up to 1/ ~ in the case that the Lagrangian state is associated with an unstable manifold of the geodesic flow. 1