Inverse scattering theory for Schr\"odinger operators with steplike potentials

TL;DR

This paper develops a comprehensive framework for the inverse scattering problem of one-dimensional Schrödinger operators with steplike potentials, crucial for solving integrable PDEs like the KdV equation.

Contribution

It provides necessary and sufficient conditions for scattering data to correspond to potentials with specific smoothness and decay properties.

Findings

Characterized scattering data for steplike potentials

Established conditions linking data to potential properties

Facilitated inverse scattering solutions for KdV

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.