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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba57-2-11,
author = {A. G. Ramm},
title = {On the Relation between the S-matrix and the Spectrum of the Interior Laplacian},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
volume = {57},
year = {2009},
pages = {181-188},
zbl = {1175.78012},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba57-2-11}
}
A. G. Ramm. On the Relation between the S-matrix and the Spectrum of the Interior Laplacian. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 57 (2009) pp. 181-188. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba57-2-11/