The Schrödinger propagator for scattering metrics. (English) Zbl 1126.58016

Hassell, Andrew; Wunsch, Jared

Ann. Math. (2) 162, No. 1, 487-523 (2005).

Summary: 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 \(\mathbb R^n\). Consider the operator \(H = \Delta + V\), 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 \(X\). 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 \(z \to X\), 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 \(X \times X^{\circ} \times \) 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 \(X\). 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.

MSC:

58J50

Spectral problems; spectral geometry; scattering theory on manifolds