Inverse scattering. III. Three dimensions, continued

This paper presents further progress in the solution of the three-dimensional inverse scattering problem for the Schrödinger equation. We prove that if the potential is in a specified class and produces no bound states, then the kernel of the generalized Marchenko equation defines a compact operator and the equation has a unique solution unless the operator has the eigenvalue 1. A partial characterization of scattering amplitudes associated with underlying local potentials without bound states is given and the potential is constructed without assuming its existence. An improved generalization of the Marchenko equation is presented for the case with bound states. The generalized Gel’fand–Levitan equation is critically reviewed.