Cantor Spectrum for CMV Matrices With Almost Periodic Verblunsky Coefficients

TL;DR

This paper proves that for certain quasi-periodic CMV matrices with small perturbations, the spectrum forms a Cantor set with all possible gaps open, extending to quantum walk models.

Contribution

It establishes the Cantor spectrum for CMV matrices with almost periodic Verblunsky coefficients and analyzes the analyticity of the spectrum boundaries.

Findings

All allowed spectral gaps are generically open.

Spectrum is a Cantor set under specified conditions.

Results apply to quantum walk models.

Abstract

We consider extended CMV matrices with analytic quasi-periodic Verblunsky coefficients with Diophantine frequency vector in the perturbatively small coupling constant regime and prove the analyticity of the tongue boundaries. As a consequence we establish that, generically, all gaps of the spectrum that are allowed by the Gap Labelling Theorem are open and hence the spectrum is a Cantor set. We also prove these results for a related class of almost periodic Verblunsky coefficients and present an application to suitable quantum walk models on the integer lattice.