Classification of integrable (beta)-equations

It is well known that the main difficulty in solving eigenvalue problems under shape deformation relates to the continuation of multiple eigenvalues of the unperturbed configuration. 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 operatorvalued functions that are of Fredholm type with index 0. We then proceed from the generalized Roucheʹs theorem to investigate the splitting problem.