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

Abstract

Consider a compact manifold with boundary M with a scattering metric g or, equivalently, an asymptotically conic manifold ( M ○ , g ) . (Euclidean R n , 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 ( h 2 Δ + V − ( λ 0 ± i 0 ) 2 ) −1 , at a non-trapping energy λ 0 > 0 , uniformly for h ∈ ( 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 − i t ( Δ / 2 + V ) , t ∈ ( 0 , t 0 ) as a quadratic Legendre distribution. We also determine the global semiclassical structure of the spectral projector, Poisson operator and scattering matrix.