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On the relation between the S-matrix and the spectrum of the interior Laplacian

Ramm A. G.

Bulletin of the Polish Academy of Sciences. Mathematics

2009

Vol. 57, no 2

181-188

The main results of this paper are: 1) a proof that a necessary condition for 1 to be an eigenvalue of the S-matrix is real analyticity of the boundary of the obstacle, 2) a short proof that if 1 is an eigenvalue of the S-matrix, then k² is an eigenvalue of the Laplacian of the interior problem, and that in this case there exists a solution to the interior Dirichlet problem for the Laplacian, which admits an analytic continuation to the whole space R³ as an entire function.