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Scattering asymptotics for Riemann surfaces. (English) Zbl 0898.58054

The authors provide a proof of the optimal polynomial lower bound for the number of resonances of a surface with hyperbolic ends. They also give Weyl asymptotics for relative scattering phase of such a surface. The authors remark that the main technical ingredient in their paper is a new proof of the Poisson formula, which also applies in the Euclidean model, and that from the point of view of scattering theory – finite volume quotients have one-dimensional infinity [one may see: J. Sjöstrand and M. Zworski, Commun. Partial Differ. Equations 17, No. 5/6, 1021-1035 (1992; Zbl 0766.35031)].
The methods used in the paper (except for the general aspects of the proof of the Poisson formula) are related to L. Guillopé and M. Zworski [J. Funct. Anal. 129, No. 2, 364-389 (1995; Zbl 0841.58063)].

MSC:

58J37 Perturbations of PDEs on manifolds; asymptotics
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
35P25 Scattering theory for PDEs
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