The Schrödinger Propagator for Scattering Metrics

• Published in: Annals of mathematics 2005-07, Vol.162 (1), p.487-523
• Main Authors: Hassell, Andrew, Wunsch, Jared
• Format: Article
• Language: English
• Subjects: Conic sections; Coordinate systems; Cotangent function; Differential operators; Distribution theory; Exact sciences and technology; Infinity; Lagrangian function; Mathematical manifolds; Mathematics; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Tangents; Vector fields
• ISSN: 0003-486X; 1939-8980
• DOI: 10.4007/annals.2005.162.487

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 Rn. Consider the operator$H=\frac{1}{2}\Delta +V$, 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 ∂ X. Assuming that g is nontrapping, we construct a global parametrix${\cal U}(z,w,t)$for the kernel of the Schrödinger propagator$U(t)=e^{-itH}$, where$z,w\in X{{}^\circ}$and t ≠ 0. The parametrix is such that the difference between U and U is smooth and rapidly decreasing both as t → 0 and as$z\rightarrow \partial X$, 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$X\times X{{}^\circ}\times {\Bbb R}_{\geq 0}$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 ≠ 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.