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Summary: Splitting of Resonant and Scattering Frequencies Under Shape
Deformation
Habib Ammari and Faouzi Triki
Centre de Mathematiques Appliquees
CNRS UMR 7641 & Ecole Polytechnique
91128 Palaiseau Cedex, France
ammari@cmapx.polytechnique.fr
triki@cmapx.polytechnique.fr
August 18, 2003
Abstract
It is well known that the main diÆculty in solving eigenvalue problems under shape de-
formation relates to the continuation of multiple eigenvalues of the unperturbed con guration.
These eigenvalues may evolve, under shape deformation, as separated, distinct eigenvalues, and
the splitting may only become apparent at high orders in their Taylor expansion. In this paper
we address the splitting problem in the evaluation of resonant and scattering frequencies of the
two-dimensional Laplacian operator under boundary variations of the domain. By using surface
potentials we show that the eigenvalues are the characteristic values of meromorphic operator-
valued functions that are of Fredholm type with index 0. We then proceed from the generalized
Rouche's theorem to investigate the splitting problem.
Keywords: eigenvalues, eigenfunctions, shape deformation, splitting, analyticity, generalized Rouche's
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