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

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 −1(h2Δ+V−2(λ0±i0)), 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, e−it(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.