Zh. Mat. Fiz. Anal. Geom. 11, 123-158 (2015) [DOI: 10.15407/mag11.02.123]
Inverse scattering theory for Schrödinger operators with steplike potentials
Iryna Egorova, Zoya Gladka, Till Luc Lange, and Gerald Teschl
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 their asymptotics. These results are important for solving the Korteweg-de Vries equation via the inverse scattering transform.