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 = cursive Greek chi^-4 dcursive Greek chi² + cursive Greek chi^-2h′ near the boundary, where cursive Greek chi is a boundary defining function and h′ is a smooth symmetric 2-cotensor which restricts to a metric h on ∂X. Let H = Δ + V, where V ∈ cursive Greek chi²C^∞ (X) is real, so V is a 'short-range' perturbation of Δ. 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 X²_b (the blowup of X² about the corner, (∂X)²). 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, i.e., unitary operators which intertwine H with a multiplication operator and determine the scattering matrix; we also give a scattering wavefront set estimate for the resolvent R(σ ± i0) applied to a distribution f.