Anderson localization for the quasi-periodic CMV matrices with Verblunsky coefficients defined by the skew-shift

TL;DR

This paper proves Anderson localization and positive Lyapunov exponents for quasi-periodic CMV matrices with Verblunsky coefficients defined by the skew-shift, extending results known for Schrödinger operators to this setting.

Contribution

It establishes Anderson localization and positivity of Lyapunov exponents for a new class of quasi-periodic CMV matrices with skew-shift Verblunsky coefficients.

Findings

Positivity of Lyapunov exponents for most frequencies

Anderson localization established for the model

Extension of Schrödinger operator results to CMV matrices

Abstract

In this paper, we study quasi-periodic CMV matrices with Verblunsky coefficients given by the skew-shift. We prove the positivity of Lyapunov exponents and Anderson localization for most frequencies, which establish the analogous results of one-dimensional Schr\"{o}dinger operators proved by Bourgain, Goldstein and Schlag.