Scattering Theory for Jacobi Operators with General Steplike Quasi-Periodic Background

TL;DR

This paper develops a comprehensive scattering theory for Jacobi operators with steplike, quasi-periodic backgrounds, enabling the unique reconstruction of operators from scattering data with finite first moments.

Contribution

It introduces a complete characterization of scattering data for Jacobi operators with steplike, quasi-periodic coefficients, advancing inverse scattering methods for such operators.

Findings

Complete characterization of scattering data for these operators

Unique solvability of inverse problem with finite first moment perturbations

Framework applicable to operators with different asymptotic backgrounds

Abstract

We develop direct and inverse scattering theory for Jacobi operators with steplike coefficients which are asymptotically close to different finite-gap quasi-periodic coefficients on different sides. 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 first moment.