Inverse Scattering Theory for Schr?dinger Operators with Steplike Potentials

摘要

We study the direct and inverse scattering problem for the one-dimensional Schr?dinger 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.