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 R-2 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 kappa 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 kappa --> 0 and obtain uniqueness of the obstacle from Schiffer's uniqueness result. Uniqueness of the inhomogeneity also follows from kappa --> 0 together with the denseness result for the linear span of products of harmonic functions.