A uniqueness theorem for the Helmholtz' equation: Penetrable media with an infinite interface

In this paper we will prove the uniqueness of a solution to Helmholtz' equation for two halfspaces of different media in $n$ dimensions. The theorem allows a finite number of bounded inhomogeneities in each half space. The surface separating the half spaces is assumed to be a cone of arbitrary cross section far away from the origin and is furthermore assumed to be smooth. We assume all space to be lossless, and in each halfspace we assume a radiation condition to be fulfilled. The boundary conditions at the interface are a general coupling in the field and its normal derivative with constant coefficients.