  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.
