Scattering with Critically-Singular and $$\delta $$-Shell Potentials

Abstract

The authors consider a scattering problem for electric potentials that have a component which is critically singular in the sense of Lebesgue spaces, and a component given by a measure supported on a compact Lipschitz hypersurface. They study direct and inverse point-source scattering under the assumptions that the potentials are real-valued and compactly supported. To solve the direct scattering problem, the authors introduce two functional spaces ---sort of Bourgain type spaces--- that allow to refine the classical resolvent estimates of Agmon and H\"ormander, and Kenig, Ruiz and Sogge. These spaces seem to be very useful to deal with the critically-singular and $\delta$-shell components of the potentials at the same time. Furthermore, these spaces and their corresponding resolvent estimates turn out to have a strong connection with the estimates for the conjugated Laplacian used in the context of the inverse Calder\'on problem. In fact, the authors derive the classical estimates by Sylvester and Uhlmann, and the more recent ones by Haberman and Tataru after some embedding properties of these new spaces. Regarding the inverse scattering problem,the authors prove uniqueness for the potentials from point-source scattering data at fix energy. To address the question of uniqueness the authors combine some of the most advanced techniques in the construction of complex geometrical optics solutions.