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Poles of diffraction at a corner. (Pôles de diffusion engendrés par un coin.) (French) Zbl 0896.35099

Astérisque. 242. Paris: Société Mathématique de France, 122 p. (1997).
The author investigates the precise existence of poles of the scattering matrix for Dirichlet problem in an exterior domain in \(\mathbb{R}^2\) with obstacles containing a conner and a trapping trajectory connected with this conner. Denote an exterior domain in \(\mathbb{R}^2\) by \(\Omega= \mathbb{R}^2\backslash Q\), where \(Q= Q_1\cup Q_2\), (\(Q_i\) are compact domains). Assume that the boundary \(\partial Q_1\) is analytic, \(\partial Q_2\) is analytic except at one point \(O\) and \(Q_i\) are strictly convex. Let \(d\) be the distance between the obstacles.
Then it is proved that for any positive integer \(N\) there exists \(C_N>0\) such that there are \(C_N'> 0\), \(M\leq N\), \((a_p,q_p)\) \((0\leq p\leq M)\) in \(\mathbb{C}\times \mathbb{Q}^+\) such that if one denotes by \((\lambda_j,p)\) \((j\in\mathbb{Z})\), the series of solutions of the equation; \(e^{2i\lambda d}\lambda^{{1\over 2}+qp}= a^{-1}_p\), which verifies \(\text{Re }\lambda_j\), \(p\sim j/d\), \(\text{Im }\lambda_j,p\sim(1/2+ q)/(2d)\log| j|\) for each \(p\), each ball with it’s center \(\lambda_j\), \(p\) contains a pole of the scattering matrix (with multiplicity) and belongs to the set \(\{\text{Im }\lambda\leq (N+1/2)/(2d)\log|\lambda|- C_N', | \lambda|> C_N\}\).

MSC:

35P25 Scattering theory for PDEs
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
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