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Polynomial bound on the distribution of poles in scattering by an obstacle. (English) Zbl 0621.35073
Journ. Équ. Dériv. Partielles, St.-Jean-De-Monts 1984, Conf. No. 3, 8 p. (1984).
Let \(\theta \subset R^ n\), \(n\geq 3\) odd be a smooth compact obstacle. In the Lax-Phillips scattering theory the scattering matrix for \(\theta\) with Dirichlet, Neumann or Robin boundary condition is meromorphic in the complex plane. Let \(\{\mu_ j\}\) be the sequence of these poles repeated according to multiplicity and arranged to have \(| \mu_ j|\) non- decreasing. In this note it is shown that there is a constant C such that \[ (*)\quad N(r)=\max \{j;\quad | \mu_ j| \leq r\}<Cr^ n+C. \] The proof is similar to that in a previous paper by the author [J. Funct. Anal. 53, 287-303 (1983; Zbl 0535.35067)] for scattering by a potential with compact support by sufficiently simplified that this rather precise growth is obtained, of the same as for the interior problem (after assistance from D. Jerison).

MSC:
35P25 Scattering theory for PDEs
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