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).


35P25 Scattering theory for PDEs


Zbl 0535.35067
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