Spectral properties of the Dirichlet-to-Neumann operator for the exterior Helmholtz problem and its applications to scattering theory

摘要

We prove that the Dirichlet-to-Neumann operator (DtN) has no spectrum in the lower half of the complex plane. We find several applications of this fact in scattering by obstacles with impedance boundary conditions. In particular, we find an upper bound for the gradient of the scattering amplitude and for the total cross section. We justify numerical approximations by providing bounds for the difference between theoretical and approximated solutions without using any a priori unknown constants.