Inverse Scattering Theory for One-Dimensional Schroedinger Operators with Steplike Periodic Potentials

TL;DR

This paper develops a comprehensive inverse scattering theory for one-dimensional Schrödinger operators with steplike periodic potentials, enabling the unique reconstruction of potentials from scattering data.

Contribution

It introduces a complete characterization of scattering data for steplike finite-gap potentials, facilitating the inverse problem's solution with finite second moment perturbations.

Findings

Complete scattering data characterization for steplike potentials

Unique solvability of inverse problem with finite second moment

Extension of inverse scattering theory to steplike finite-gap cases

Abstract

We develop direct and inverse scattering theory for one-dimensional Schroedinger operators with steplike potentials which are asymptotically close to different finite-gap periodic potentials on different half-axes. We give a complete characterization of the scattering data, which allow unique solvability of the inverse scattering problem in the class of perturbations with finite second moment.