A Uniqueness Theorem for an Inverse Scattering Problem in an Exterior Domain

The scattering of time-harmonic acoustic waves by a simply connected, sound-soft obstacle in an inhomogeneous medium in R2 is considered. We prove that the Cauchy data of the scattered waves on a large circle for all incident waves and for an interval of wave numbers κ uniquely determine the obstacle and the inhomogeneity. To this end we show that products of harmonic functions satisfying a Dirichlet condition on the interior boundary of an annular plane domain are complete. Then, we use the limit κ → 0 and obtain uniqueness of the obstacle from Schiffer's uniqueness result. Uniqueness of the inhomogeneity also follows from κ → 0 together with the denseness result for the linear span of products of harmonic functions.