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The spectral projections and the resolvent for scattering metrics. (English) Zbl 0981.58025

{Authors’ 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 \(g=x^{-4}dx^2 +x^{-2}h^\prime\) near the boundary, where \(x\) is a boundary defining function and \(h^\prime\) 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, so \(V\) 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 \(X_b^2\) (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 \(H\), 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 \(R(\sigma\pm i0)\) applied to a distribution \(f\).

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
58J40 Pseudodifferential and Fourier integral operators on manifolds
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References:

[1] A. Hassell,Distorted plane waves for the 3 body Schrödinger operator, Geom. Funct. Anal., to appear. · Zbl 0953.35122
[2] A. Hassell,Scattering matrices for the quantum N-body problem, Trans. Amer. Math. Soc., to appear. · Zbl 0942.35125
[3] A. Hassell and A. Vasy,Symbolic functional calculus and N-body resolvent estimates, J. Funct. Anal., to appear. · Zbl 0960.58025
[4] Herbst, I.; Skibsted, E., Free channel Fourier transform in the long range N body problem, J. Analyse Math., 65, 297-332 (1995) · Zbl 0866.70008 · doi:10.1007/BF02788775
[5] Hörmander, L., Fourier integral operators, I, Acta Math., 127, 79-183 (1971) · Zbl 0212.46601 · doi:10.1007/BF02392052
[6] Jensen, A., Propagation estimates for Schrödinger-type operators, Trans. Amer. Math. Soc., 291, 129-144 (1985) · Zbl 0577.35089 · doi:10.2307/1999899
[7] R. Mazzeo and R. B. Melrose,Pseudodifferential operators on manifolds with fibred boundaries, preprint. · Zbl 1125.58304
[8] Melrose, R. B., Calculus of conormal distributions on manifolds with corners, Internat. Math. Res. Notices, 3, 51-61 (1992) · Zbl 0754.58035 · doi:10.1155/S1073792892000060
[9] Melrose, R. B., Spectral and Scattering Theory for the Laplacian on Asymptotically Euclidean Spaces (1994), New York: Marcel Dekker, New York · Zbl 0837.35107
[10] Melrose, R. B.; Uhlmann, G., Lagrangian intersection and the Cauchy problem, Comm. Pure Appl. Math., 32, 483-519 (1979) · Zbl 0396.58006 · doi:10.1002/cpa.3160320403
[11] Melrose, R. B.; Zworski, M., Scattering metrics and geodesic flow at infinity, Invent. Math., 124, 389-436 (1996) · Zbl 0855.58058 · doi:10.1007/s002220050058
[12] Reed, M.; Simon, B., Methods of Modern Mathematical Physics III: Scattering Theory (1979), New York: Academic Press, New York · Zbl 0405.47007
[13] Taylor, M. E., Partial Differential Equations (1996), Berlin: Springer, Berlin · Zbl 0869.35002
[14] A. Vasy,Propagation of singularities in three-body scattering, Astérisque, to appear. · Zbl 0941.35001
[15] A. Vasy,Propagation of singularities in three-body scattering, PhD thesis, Massachusetts Institute of Technology, 1997. · Zbl 0941.35001
[16] Vasy, A., Geometric scattering theory for long-range potentials and metrics, Internat. Math. Res. Notices, 6, 285-315 (1998) · Zbl 0922.58085 · doi:10.1155/S1073792898000208
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