Geometric scattering theory for long-range potentials and metrics. (English) Zbl 0922.58085

Let \(X\) be a compact manifold with boundary \(\partial X\) and let \(\kappa\) be a boundary defining function. The metric \(g\) on \(X\) is called a long range scattering metric if it has the form \(g=a{d\kappa^2\over\kappa^4}+{h\over x^2}\) where \(a-1\in\kappa C^\infty(X)\) and where \(h\) is a smooth symmetric \(2\) cotensor on \(X\) restricting to a metric on \(\partial X\). Let \(\Delta\) be the Laplacian of \(g\) and let \(H=\Delta+V\) where \(V\) is a suitably chosen second order operator. For example, one could take \(V\in\kappa C^\infty(X)\) (such a \(V\) is long range); if \(V\in \kappa^2 C^\infty(X)\), then \(V\) is short range. One of the main results of the paper is:
Theorem. For \(\lambda>0\), the scattering matrix \(S(\lambda)\) is a Fourier integral operator whose canonical relation is given by the (forward) geodesic flow of \(h\) on \(\partial X\) at distance \(\pi\); the imaginary part of the order of \(S(\lambda)\) varies and depends on \(\alpha_{\pm}\).
This theorem generalizes previous results by the author, by Joshi, by Melrose, and by Zworski [R. Melrose and M. Zworski, Invent. Math. 124, No. 1-3, 389-436 (1996; Zbl 0855.58058), M. S. Joshi and A. Sa Barreto, Commun. Math. Phys. 193, No. 1, 197-208 (1998), and A. Vasy, J. Funct. Anal. 148, No. 1, 170-184 (1997; Zbl 0884.35110)]. The author also computes the principal symbol of \(S(\lambda)\) and describes the structure of the Poisson operator.
Reviewer: P.Gilkey (Eugene)


58J50 Spectral problems; spectral geometry; scattering theory on manifolds
35P25 Scattering theory for PDEs
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