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


35P25 Scattering theory for PDEs
35L05 Wave equation
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