INVERSE SCATTERING FOR SINGULAR POTENTIALS IN TWO DIMENSIONS

We consider the Schrodinger equation for a compactly supported potential having jump type singularities at a subdomain of R2 . We prove that knowledge of the scattering amplitude at a fixed energy, determines the location of the singularity as well as the jump across the curve of discontinuity. This result follows from a similar result for the Dirichlet to Neumann map associated to the Schrodinger equation for a compactly supported potential with the same type of singularities. 1. Introduction and statement of the results In this paper we consider the Schrodinger equation for a compactly supported potential, q , having jump type singularities at a subdomain of K2. We prove that knowledge of the scattering amplitude at a fixed energy, Ao , determines the location of the singularity as well as the jump across the curve of discontinuity. This problem is reduced to the study of the Dirichlet to Neumann map for the Schrodinger operator -A + q - X$ in a bounded domain of R2 . (For the application considered here it is enough to consider the domain to be a ball containing the support of q .) We prove that in dimension two the Dirichlet to Neumann map for the Schrodinger operator -A + q-X^ determines uniquely the location of the singularity of q as well as its jump across the curve of discontinuity. The scattering amplitude of a potential q £ L°°(Rn) with compact support is defined via the outgoing eigenfunctions. Namely, MX £ R\0, co £ S"~x , there exists i//+ (X, x, co), solution of