Scattering theory for bulk and surface waves in an impedance half-space, II: local perturbations, the scattering matrix and an inverse problem

Abstract

We study the Robin, or impedance, Laplacian on a half-space of dimension $n \ge 3$ which is perturbed inside a fixed ball. The boundary may be deformed, and obstacles may be present in the interior or attached to the boundary, in any combination and with no smallness assumption. Outside the ball the configuration is the flat impedance half-space of Part I, with constant impedance $α> 0$, where a bulk channel and a surface channel propagate at every positive energy and only the surface channel propagates on the band $(-α^2,0)$. We prove a limiting absorption principle, valid at every positive energy and outside a discrete set of embedded eigenvalues in the band. We construct the generalized Fourier transform and prove its completeness by a stationary argument which constructs no wave operator and reduces completeness to the uniqueness of the scattering solution at each energy. We then obtain the scattering matrix, its unitarity and a two-channel optical theorem. The main result is a uniqueness theorem for the inverse problem. Under two geometric conditions on the pair, the scattering amplitude at a single positive energy determines the perturbation completely, and the argument does not distinguish a deformation of the boundary from an obstacle. It follows that no nontrivial perturbation satisfying the same conditions is invisible at any positive energy. On the surface band the corresponding statement is reduced to one explicitly stated question about the exterior Robin problem.