TAILIEUCHUNG - Đề tài "The Schr¨odinger propagator for scattering metrics "

Let g be a scattering metric on a compact manifold X with boundary, ., a smooth metric giving the interior X ◦ 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 = 1 ∆ + V , where ∆ is the positive Laplacian with respect to g and V is a 2 smooth real-valued function on X vanishing to second order at ∂X. Assuming that g is nontrapping, we construct a global parametrix U(z, w, t) for the kernel. | Annals of Mathematics The Schr odinger propagator for scattering metrics By Andrew Hassell and Jared Wunsch Annals of Mathematics 162 2005 487 523 The Schrodinger propagator for scattering metrics By Andrew Hassell and Jared Wunsch Abstract Let g be a scattering metric on a compact manifold X with boundary . a smooth metric giving the interior X the structure of a complete Rieman-nian manifold with asymptotically conic ends. An example is any compactly supported perturbation of the standard metric on Rra. Consider the operator H 1A V where A is the positive Laplacian with respect to g and V is a smooth real-valued function on X vanishing to second order at dX. 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 E X 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 dX uniformly for w on compact subsets of X . 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 2tU t . Using the parametrix we show that W t belongs to a class of Legendre distributions on X X X X Ryo 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 dX. 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. 1. Introduction Let X g be a scattering manifold of dimension n. Thus X is a compact .

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