The Schrödinger 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 $X^\circ$ the structure of a complete Riemannian manifold with asymptotically conic ends. An example is any compactly supported perturbation of the standard metric on $\mathrm{R}^n$. Consider the operator $H = \frac{1}{2} \Delta + V$, 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 $\partial X$. Assuming that $g$ is nontrapping, we construct a global parametrix $\mathcal{U}(z, w,t)$ for the kernel of the Schrödinger propagator $U(t) = e^{-i t H}$, where $z, w \in X^{\circ}$ and $t \neq 0$. The parametrix is such that the difference between $\mathcal{U}$ and $U$ is smooth and rapidly decreasing both as $t \to 0$ and as $z \to \partial X$, uniformly for $w$ on compact subsets of $X^{\circ}$. Let $r = x^{-1}$, where $x$ is a boundary defining function for $X$, be an asymptotic radial variable, and let $W(t)$ be the kernel $e^{-ir^2/2t}U(t)$. Using the parametrix, we show that $W(t)$ belongs to a class of ‘Legendre distributions’ on $X \times X^{\circ} \times \mathbb{R}_{\geq 0}$ previously considered by Hassell-Vasy. When the metric is trapping, then the parametrix construction goes through microlocally in the nontrapping part of the phase space.

We apply this result to determine the singularities of $U(t) f$, for any tempered distribution $f$ and for any fixed $t \neq 0$, in terms of the oscillation of $f$ near $\partial X$. If the metric is nontrapping then we precisely determine the wavefront set of $U(t) f$, and hence also precisely determine its singular support. More generally, we are able to determine the wavefront set of $U(t) f$ for $t> 0$, resp. $t < 0$ on the non-backward-trapped, resp. non-forward-trapped subset of the phase space. This generalizes results of Craig-Kappeler-Strauss and Wunsch.

Authors

Andrew Hassell

Mathematical Sciences Institute
Australian National University
Canberra 0200 ACT
Australia

Jared Wunsch

Department of Mathematics
Northwestern University
Evanston, IL 60208
United States