  We study solutions of the time dependent Schr\"odinger equation on Riemannian
manifolds with oscillatory initial conditions given by Lagrangian states.
Semiclassical approximations describe these solutions for small h (where h is
the semiclassical parameter), but their accuracy for large times is in general
only understood up to the Ehrenfest time T ~ ln(1/h), 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 T ~ h^(-1/2) in the
case that the Lagrangian state is associated with an unstable manifold of the
geodesic flow.
