The spectral projections and the resolvent for scattering metrics
 Andrew Hassell,
 András Vasy
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In this paper, we consider a compact manifold with boundaryX equipped with a scattering metricg as defined by Melrose [9]. That is,g is a Riemannian metric in the interior ofX that can be brought to the formg=x ^{−4} dx^{2}+x^{−2 h’} near the boundary, wherex is a boundary defining function andh’ is a smooth symmetric 2cotensor which restricts to a metrich on ϖX. LetH=Δ+V, whereV∈x ^{2}C^{∞} (X) is real, soV is a ‘shortrange’ perturbation of Δ. Melrose and Zworski started a detailed analysis of various operators associated toH in [11] and showed that the scattering matrix ofH is a Fourier integral operator associated to the geodesic flow ofh on ϖX at distance π and that the kernel of the Poisson operator is a Legendre distribution onX×ϖX 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 bstretched productX _{b} ^{2} (the blowup ofX ^{2} about the corner, (ϖX)^{2}). The structure of the resolvent is only slightly more complicated.
As applications of our results, we show that there are ‘distorted Fourier transforms’ forH, i.e., unitary operators which intertwineH with a multiplication operator and determine the scattering matrix; we also give a scattering wavefront set estimate for the resolventR(σ±i0) applied to a distributionf.
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 Title
 The spectral projections and the resolvent for scattering metrics
 Journal

Journal d’Analyse Mathématique
Volume 79, Issue 1 , pp 241298
 Cover Date
 19991201
 DOI
 10.1007/BF02788243
 Print ISSN
 00217670
 Online ISSN
 15658538
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Authors

 Andrew Hassell ^{(1)}
 András Vasy ^{(2)}
 Author Affiliations

 1. Centre for Mathematics and Its Applications, Australian National University, 0200, Canberra, ACT, Australia
 2. Department of Mathematics, University of California, 94720, Berkeley, CA, USA