Scattering theory for CMV matrices: uniqueness, Helson--Szeg\H{o} and Strong Szeg\H{O} theorems

TL;DR

This paper develops a scattering theory for CMV matrices, establishing conditions for the uniqueness of inverse problems and connecting these to classical theorems like Helson--Szeg ext{o} and Szeg ext{o}.

Contribution

It introduces a scattering framework for CMV matrices and provides new criteria for the uniqueness of inverse spectral problems, linking them to classical harmonic analysis theorems.

Findings

Characterization of Verblunsky parameters and spectral measures.

Necessary and sufficient conditions for inverse problem uniqueness.

Connections between scattering operators and classical theorems.

Abstract

We develop a scattering theory for CMV matrices, similar to the Faddeev--Marchenko theory. A necessary and sufficient condition is obtained for the uniqueness of the solution of the inverse scattering problem. We also obtain two sufficient conditions for the uniqueness, which are connected with the Helson--Szeg\H o and the Strong Szeg\H o theorems. The first condition is given in terms of the boundedness of a transformation operator associated to the CMV matrix. In the second case this operator has a determinant. In both cases we characterize Verblunsky parameters of the CMV matrices, corresponding spectral measures and scattering functions.