On the Relation between the $S$-matrix and the Spectrum of the Interior Laplacian

Tom 57 / 2009

A. G. Ramm Bulletin Polish Acad. Sci. Math. 57 (2009), 181-188 MSC: 78A45, 35J25. DOI: 10.4064/ba57-2-11


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 $k^2$ 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 $\mathbb R^3$ as an entire function.


  • A. G. RammMathematics Department
    Kansas State University
    Manhattan, KS 66506-2602, U.S.A.

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