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The semiclassical resolvent and the propagator for nontrapping scattering metrics | | Andrew Hassell
; Jared Wunsch
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23 Jun 2006 | | Subject: | Analysis of PDEs; Spectral Theory | | Abstract: | Consider a compact manifold with boundary $M$ with a scattering metric $g$ or, equivalently, an asymptotically conic manifold $(M^circ, g)$. (Euclidean $mathbb{R}^n$, with a compactly supported metric perturbation, is an example of such a space.) Let $Delta$ be the positive Laplacian on $(M,g)$, and $V$ a smooth potential on $M$ which decays to second order at infinity. In this paper we construct the kernel of the operator $(h^2 Delta + V - (lambda_0 pm i0)^2)^{-1}$, at a nontrapping energy $lambda_0 > 0$, uniformly for $h in (0, h_0)$, $h_0 > 0$ small, within a class of Legendre distributions on manifolds with codimension three corners. Using this we construct the kernel of the propagator, $e^{-it(Delta/2 + V)}$, $t in (0, t_0)$ as a quadratic Legendre distribution. We also determine the global semiclassical structure of the spectral projector, Poisson operator and scattering matrix. | | Source: | arXiv, math/0606606 | | Services: | Forum | Review | PDF | Favorites | |
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