## 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)

### MSC:

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

### Citations:

Zbl 0855.58058; Zbl 0884.35110
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