We discuss the form of the propagator $U(t)$ for the time-dependent
Schr\"odinger equation on an asyptotically Euclidean, or, more generally,
asymptotically conic, manifold with no trapped geodesics. In the asymptotically
Euclidean case, if $\chi \in \mathcal{C}_0^\infty$, and with $\mathcal{F}$
denoting Fourier transform, $\mathcal{F}\circ e^{-ir^2/2t} U(t) \chi$ is a
Fourier integral operator for $t\neq 0.$ The canonical relation of this
operator is a ``sojourn relation'' associated to the long-time geodesic flow.
This description of the propagator follows from its more precise
characterization as a ``scattering fibered Legendrian,'' given by the authors
in a previous paper and sketched here.
A corollary is a propagation of singularities theorem that permits a complete
description of the wavefront set of a solution to the Schr\"odinger equation,
restricted to any fixed nonzero time, in terms of the oscillatory behavior of
its initial data. We discuss two examples which illustrate some extremes of
this propagation behavior.