Analytical characterization and numerical approximation of interior eigenvalues for impenetrable scatterers from far fields

We characterize the interior eigenvalues of a class of impenetrable, non-absorbing scattering objects from the spectra of the corresponding far field operators for a continuum of wave numbers. Our proof simplifies arguments from the original proof for Dirichlet scattering objects given in Eckmann and Pillet (1995 Commun. Math. Phys. 170 283–313) and furthermore extends to the cases of Neumann and Robin scattering objects. Further, the analytical characterization of interior eigenvalues of a scatterer can be exploited numerically. We present an algorithm that approximates interior eigenvalues from far field data without knowing the scattering object, we give several numerical examples for different scatterers and sound-hard as well as sound-soft boundary conditions, and we finally show through numerical examples that this algorithm remains stable under noise. (paper)