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 (h² Δ + V - (λ₀ ± i 0)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, e^{- i t (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.