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

Abstract

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 of the conclusion stating 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.