Reflectionless CMV matrices and scattering theory

TL;DR

This paper investigates reflectionless CMV matrices through scattering theory, deriving explicit formulas for the scattering matrix and establishing a key characterization of reflectionless matrices via the scattering matrix's structure.

Contribution

It introduces a novel approach to analyze reflectionless CMV matrices by linking their reflectionless property to the off-diagonal form of the scattering matrix, paralleling results for Jacobi matrices.

Findings

A CMV matrix is reflectionless iff the scattering matrix is off-diagonal.

Explicit formulas for the scattering matrix are derived.

Decoupling a CMV matrix into two half-line operators is achieved by changing a single Verblunsky coefficient.

Abstract

Reflectionless CMV matrices are studied using scattering theory. By changing a single Verblunsky coefficient a full-line CMV matrix can be decoupled and written as the sum of two half-line operators. Explicit formulas for the scattering matrix associated to the coupled and decoupled operators are derived. In particular, it is shown that a CMV matrix is reflectionless iff the scattering matrix is off-diagonal which in turn provides a short proof of an important result of [Breuer-Ryckman-Simon]. These developments parallel those recently obtained for Jacobi matrices.