## On the poles of the scattering matrix for two strictly convex obstacles.(English)Zbl 0561.35060

Denote by S($$\sigma)$$ the scattering matrix for the Dirichlet initial boundary value problem for the wave equation in the exterior domain $${\mathcal D}$$ of an open, bounded set $${\mathcal O}$$ with sufficiently smooth boundary $$\Gamma$$. S($$\sigma)$$ is a unitary mapping from $$L_ 2(S^ 2)$$ onto $$L_ 2(S^ 2)$$ for any $$\sigma\in {\mathbb{R}}$$ and determines the scattering obstacle uniquely. The author considers the question of relating geometrical properties of the obstacle $${\mathcal O}$$ and the distribution of poles of the meromorphic operator family S(z), $$z\in {\mathbb{C}}$$. The main result of the paper provides information of the position of singularities of the scattering matrix in a particular trapping case: $${\mathcal O}$$ consists of two disjoint strictly convex obstacles. The result is based on a refinement of a parametrix construction used in previous work of the author. The proofs are given in detail.
Reviewer: R.Picard

### MSC:

 35P25 Scattering theory for PDEs 35L05 Wave equation
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