Elsevier

Advances in Mathematics

Volume 217, Issue 2, 30 January 2008, Pages 586-682
Advances in Mathematics

The semiclassical resolvent and the propagator for non-trapping scattering metrics

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Abstract

Consider a compact manifold with boundary M with a scattering metric g or, equivalently, an asymptotically conic manifold (M,g). (Euclidean Rn, with a compactly supported metric perturbation, is an example of such a space.) Let Δ 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 (h2Δ+V(λ0±i0)2)−1, at a non-trapping energy λ0>0, uniformly for h(0,h0), h0>0 small, within a class of Legendre distributions on manifolds with codimension three corners. Using this we construct the kernel of the propagator, eit(Δ/2+V), t(0,t0) as a quadratic Legendre distribution. We also determine the global semiclassical structure of the spectral projector, Poisson operator and scattering matrix.

Keywords

Resolvent
Semiclassical
Scattering matrix
Propagator
Legendrian
Scattering manifold

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