2015•Zurnal matematiceskoj fiziki analiza geometriiOpen access

Inverse Scattering Theory for Schrodinger Operators with Steplike Potentials

Iryna Egorova, Zoya Gladka, Till Luc Lange, Gerald Teschl

Abstract

We study the direct and inverse scattering problem for the one-dimensional Schr\"odinger 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.

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Key concepts: Inverse scattering transform, Inverse scattering problem, Quantum inverse scattering method, Scattering, Smoothness, Korteweg–de Vries equation, Schrödinger equation, Scattering theory