The Schrodinger propagator for scattering metrics

• 成果类型: Article
• 署名作者: Hassell, A; Wunsch, J
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2005.162.487
• 发表日期: 2005
• 页码: 487-523
• 关键词: equation; SINGULARITIES; resolvent; Operators; SPACES

摘要

Let g be a scattering metric on a compact manifold X with boundary, i.e., a smooth metric giving the interior X degrees the structure of a complete Riemannian manifold with asymptotically conic ends. An example is any compactly supported perturbation of the standard metric on R-n. Consider the operator H = 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 derivative X. Assuming that g is nontrapping, we construct a global parametrix U (z, w, t) for the kernel of the Schrodinger propagator U(t) = e(-itH), where z, w is an element of X degrees and t not equal 0. The parametrix is such that the difference between U and U is smooth and rapidly decreasing both as t -> 0 and as z -> partial derivative X, uniformly for w on compact subsets of X degrees. 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(-ir2/2t)U(t). Using the parametrix, we show that W(t) belongs to a class of 'Legendre distributions' on X x X degrees x R->= 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 not equal 0, in terms of the oscillation of f near partial derivative 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.