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

Zbl 0535.35067
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