The Schrödinger propagator for scattering metrics

Andrew Hassell¹ ; Jared Wunsch²

¹ Mathematical Sciences Institute, Australian National University, Canberra 0200 ACT, Australia
² Department of Mathematics, Northwestern University, Evanston, IL 60208, United States

Annals of mathematics, Tome 162 (2005) no. 1, pp. 487-523

Résumé

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.

Détail

DOI : 10.4007/annals.2005.162.487