The spectral projections and the resolvent for scattering metrics
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
In this paper, we consider a compact manifold with boundaryX equipped with a scattering metricg as defined by Melrose . That is,g is a Riemannian metric in the interior ofX that can be brought to the formg=x −4 dx2+x−2 h’ near the boundary, wherex is a boundary defining function andh’ is a smooth symmetric 2-cotensor which restricts to a metrich on ϖX. LetH=Δ+V, whereV∈x 2C∞ (X) is real, soV is a ‘short-range’ perturbation of Δ. Melrose and Zworski started a detailed analysis of various operators associated toH in  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 b-stretched 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.
- A. Hassell,Distorted plane waves for the 3 body Schrödinger operator, Geom. Funct. Anal., to appear.
- A. Hassell,Scattering matrices for the quantum N-body problem, Trans. Amer. Math. Soc., to appear.
- A. Hassell and A. Vasy,Symbolic functional calculus and N-body resolvent estimates, J. Funct. Anal., to appear.
- I. Herbst and E. Skibsted,Free channel Fourier transform in the long range N body problem, J. Analyse Math.65 (1995), 297–332. CrossRef
- L. Hörmander,Fourier integral operators, I, Acta Math.127 (1971), 79–183. CrossRef
- A. Jensen,Propagation estimates for Schrödinger-type operators, Trans. Amer. Math. Soc.291 (1985), 129–144. CrossRef
- R. Mazzeo and R. B. Melrose,Pseudodifferential operators on manifolds with fibred boundaries, preprint.
- R. B. Melrose,Calculus of conormal distributions on manifolds with corners, Internat. Math. Res. Notices, No.3 (1992), 51–61. CrossRef
- R. B. Melrose,Spectral and Scattering Theory for the Laplacian on Asymptotically Euclidean Spaces, Marcel Dekker, New York, 1994.
- R. B. Melrose and G. Uhlmann,Lagrangian intersection and the Cauchy problem, Comm. Pure Appl. Math.32 (1979), 483–519. CrossRef
- R. B. Melrose and M. Zworski,Scattering metrics and geodesic flow at infinity, Invent. Math.124 (1996), 389–436. CrossRef
- M. Reed and B. Simon,Methods of Modern Mathematical Physics III: Scattering Theory, Academic Press, New York, 1979.
- M. E. Taylor,Partial Differential Equations, Volume II, Springer, Berlin, 1996.
- A. Vasy,Propagation of singularities in three-body scattering, Astérisque, to appear.
- A. Vasy,Propagation of singularities in three-body scattering, PhD thesis, Massachusetts Institute of Technology, 1997.
- A. Vasy,Geometric scattering theory for long-range potentials and metrics, Internat. Math. Res. Notices No.6 (1998), 285–315. CrossRef
- The spectral projections and the resolvent for scattering metrics
Journal d’Analyse Mathématique
Volume 79, Issue 1 , pp 241-298
- Cover Date
- Print ISSN
- Online ISSN
- Additional Links