The Schroedinger propagator for scattering metrics

Abstract

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 $H = \half Δ+ V$, where $Δ$ 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.