The semiclassical resolvent and the propagator for nontrapping scattering metrics

Details The semiclassical resolvent and the propagator for nontrapping scattering metrics The semiclassical resolvent and the propagator for nontrapping scattering metrics

2006-06-23

arxiv.org/abs/math/0606606v1

Mathematics

Analysis of PDEs

Scientific paper

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.

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