Scattering theory with finite-gap backgrounds: Transformation operators and characteristic properties of scattering data

TL;DR

This paper develops a comprehensive scattering theory for Jacobi operators with finite-gap backgrounds, providing conditions for scattering data in steplike, quasi-periodic settings with finite-moment perturbations.

Contribution

It introduces new direct and inverse scattering methods for Jacobi operators with steplike finite-gap backgrounds, extending previous theories to more general perturbations.

Findings

Necessary and sufficient conditions for scattering data with finite second or higher moments.

Characterization of scattering data for operators with steplike, finite-gap asymptotics.

Framework applicable to a broad class of perturbations in discrete spectral problems.

Abstract

We develop direct and inverse scattering theory for Jacobi operators (doubly infinite second order difference operators) with steplike coefficients which are asymptotically close to different finite-gap quasi-periodic coefficients on different sides. We give necessary and sufficient conditions for the scattering data in the case of perturbations with finite second (or higher) moment.