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Scattering by two convex bodies. (English) Zbl 0781.35046

Sémin. Équ. Dériv. Partielles, Éc. Polytech., Cent. Math., Palaiseau 1991-1992, No.XIII, 9 p. (1992).
Consider the boundary value problem (1) \(\Delta u+\mu^ 2 u=0\) in \(\Omega\), \(u=g\) on \(\Gamma\) for \(g\in C^ \infty(\Gamma)\), where \(\Omega\) is the outer domain of two smooth bounded obstacles in \(\mathbb{R}_ 2\) and \(\Gamma\) is its boundary. Then, for \(\text{Im }\mu<0\), (1) has a unique solution \(u\in L_ 2(\Omega)\). Let \(u(x)= (U(\mu)g)(x)\) then by the regularity theory for elliptic problems \(U(\mu)\) is a holomorphic function in \(\{\mu;\text{ Im }\mu<0\}\) with values in the set of continuous operators from \(C^ \infty(\Gamma)\) into \(C^ \infty(\overline\Omega)\).
In this paper, the author gives a theorem concerning the analytic continuation of \(U(\mu)\) in \(\{\mu;\text{ Im }\mu\geq 0\}\). The investigation is motivated by the problem in scattering theory to know relationships between the geometry of obstacles and the distribution of poles of scattering matrices.

MSC:

35P25 Scattering theory for PDEs
30B60 Completeness problems, closure of a system of functions of one complex variable
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
78A45 Diffraction, scattering
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