Inverse Scattering Theory for Schrödinger Operators with Steplike Potentials - Egorova, I.; Gladka, Z.; Lange, T.L.; Teschl, G. (Журнал математической физики, анализа, геометрии, 2015)
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.