Uniqueness for the inverse fixed angle scattering problem

Abstract

AbstractWe present a uniqueness result in dimensions 3 for the inverse fixed angle scattering problem associated to the Schrödinger operator {-\Delta+q}, where q is a small real-valued potential with compact support in the Sobolev space {W^{\beta,2}}, with {\beta>0.} This result improves the known result [P. Stefanov, Generic uniqueness for two inverse problems in potential scattering, Comm. Partial Differential Equations 17 1992, 55–68], in the sense that almost no regularity is required for the potential. The uniqueness result still holds in dimension 4, but for more regular potentials in {W^{\beta,2}}, with {\beta>2/3}. The proof is a consequence of the reconstruction method presented in our previous work, [J. A. Barceló, C. Castro, T. Luque and M. C. Vilela, A new convergent algorithm to approximate potentials from fixed angle scattering data, SIAM J. Appl. Math. 78 2018, 2714–2736].