Inverse Scattering Theory for Schrödinger Operators with Steplike Potentials

We study the direct and inverse scattering problem for the one-dimensional Schrödinger equation with steplike potentials. We give necessary and sufficient conditions for the scattering data to correspond to a potential with prescribed smoothness and prescribed decay to its asymptotics. Our results generalize all previous known results and are important for solving the Korteweg–de Vries equation via the inverse scattering transform.