  Let g be a scattering metric on a compact manifold X with boundary, i.e., a
smooth metric giving the interior of X the structure of a complete Riemannian
manifold with asymptotically conic ends. An example is any compactly supported
perturbation of the standard metric on Euclidean space. Consider the operator
$H = \half \Delta + V$, where $\Delta$ is the positive Laplacian with respect
to g and V is a smooth real-valued function on X vanishing to second order at
the boundary. Assuming that g is non-trapping, we construct a global parametrix
for the kernel of the Schroedinger propagator $U(t) = e^{-itH}$ and use this to
show that the kernel of U(t) is, up to an explicit quadratic oscillatory
factor, a class of `Legendre distributions' on $X \times X^{\circ} \times
\halfline$ previously considered by Hassell-Vasy. When the metric is trapping,
then the parametrix construction goes through microlocally in the non-trapping
part of the phase space.
  We apply this result to obtain a microlocal characterization of the
singularities of $U(t) f$, for any tempered distribution $f$ and any fixed $t
\neq 0$, in terms of the oscillation of f near the boundary of X. If the metric
is non-trapping, then we obtain a complete characterization; more generally we
need to assume that f is microsupported in the nontrapping part of the phase
space. This generalizes results of Craig-Kappeler-Strauss and Wunsch.
