Time-dependent scattering theory on manifolds

Based on our previous study [IS3] on the stationary scattering theory for the Schrodinger operator on a manifold possessing an escape function we complete our investigation by doing the time-dependent counterpart. A particular class of examples are manifolds with Euclidean and/or hyperbolic ends, possibly with unbounded and non-smooth obstacles. As an application we resolve a conjecture of [HPW] on cross-ends transmissions in its natural and strong form within the time-dependent framework.