The Point Source Inverse Back-Scattering Problem

We consider the inverse problem of recovering a potential by measuring the response at a point to a source located at the same point and then varying the point on the surface of a sphere. This is a similar to the inverse back-scattering problem. We show that if the angular derivatives of the difference of two potentials having the same data is controlled by the L^2 norm of the difference of the potentials they must be equal. In particular this shows injectivity of the inverse problem for radial potentials.