The Spectral Projections and the Resolvent for Scattering Metrics

In this paper we consider a compact manifold with boundary X equipped with a scattering metric g as defined by Melrose [9]. That is, g is a Riemannian metric in the interior of X that can be brought to the form g = x dx + xh near the boundary, where x is a boundary defining function and h is a smooth symmetric 2-cotensor which restricts to a metric h on ∂X. Let H = ∆ + V where V ∈ xC(X) is real, so V is a ‘shortrange’ perturbation of ∆. Melrose and Zworski started a detailed analysis of various operators associated to H in [11] and showed that the scattering matrix of H is a Fourier integral operator associated to the geodesic flow of h on ∂X at distance π and that the kernel of the Poisson operator is a Legendre distribution on X×∂X associated to an intersecting pair with conic points. In this paper we describe the kernel of the spectral projections and the resolvent, R(σ± i0), on the positive real axis. We define a class of Legendre distributions on certain types of manifolds with corners, and show that the kernel of the spectral projection is a Legendre distribution associated to a conic pair on the b-stretched product X b (the blowup of X about the corner, (∂X)). The structure of the resolvent is only slightly more complicated. As applications of our results we show that there are ‘distorted Fourier transforms’ for H, ie, unitary operators which intertwine H with a multiplication operator and determine the scattering matrix; and give a scattering wavefront set estimate for the resolvent R(σ ± i0) applied to a distribution f .