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Research Report MRR98-061

The spectral projections and the resolvent for scattering metrics

Andrew Hassell and András Vasy

Abstract: In this paper we consider a compact manifold with boundary X equipped with a scattering metric g as defined by Melrose. That is, g is a Riemannian metric in the interior of X that can be brought to the form $g=x^{-4}\,dx^2+x^{-2}h'$ 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 $\partial X$ . Let $H=\Delta+V$ where $V\in x^2 C^\infty (X)$ is real, soV is a `short-range' perturbation of $\Delta$ . Melrose and Zworski started a detailed analysis of various operators associated to H and showed that the scattering matrix of H is a Fourier integral operator associated to the geodesic flow of h on $\partial X$ at distance $\pi$ and that the kernel of the Poisson operator is a Legendre distribution on $X\times \partial X$ associated to an intersecting pair with conic points. In this paper we describe the kernel of the spectral projections and the resolvent, $R(\sigma \pm i0)$ , 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 (the blowup of $X^2$ about the corner, $(\partial X)^2$ ). 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, ie, unitary operators which intertwine H with a multiplication operator and determine the scattering matrix; and give a scattering wavefront set estimatefor the resolvent $R(\sigma \pm i0)$ applied to a distribution f.

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