On Scattering for CMV Matrices

TL;DR

This paper develops a scattering representation for CMV matrices using the Adamjan-Arov model space, explicitly computes the basis and Verblunski coefficients via Nehari interpolation, and explores their asymptotic behavior and spectral relations.

Contribution

It introduces a scattering representation for CMV matrices, explicitly computes the basis and coefficients, and links scattering and spectral representations.

Findings

Verblunski coefficients tend to zero asymptotically

Explicit formulas for basis and coefficients via Nehari interpolation

Established relations between basis and wandering subspaces

Abstract

Adamjan-Arov (Lax--Phillips) model space is considered as a scattering representation space for a CMV matrix in context of an extended Marchenko--Faddeev scattering theory. That is, there exists a basis in which the multiplication by independent variable is a CMV matrix. This basis as well as Verblunski coefficients are computed explicitly in terms of Nehari interpolation. Asymptotically the Verblynski coefficients go to zero. Moreover, relations between the basis and wandering subspaces are established. Transformation from scattering representation to spectral representation is given.