The spectrum of the multi-frequency quasi-periodic CMV matrices contains intervals

TL;DR

This paper studies the spectral properties of multi-frequency quasi-periodic CMV matrices, demonstrating that their spectra include intervals on the unit circle under certain conditions, advancing understanding of their spectral structure.

Contribution

It proves that the spectrum of these matrices contains intervals on the unit circle in the positive Lyapunov exponent regime with Diophantine frequencies, a new insight into their spectral nature.

Findings

Spectra contain intervals on the unit circle.

Results hold under positive Lyapunov exponent conditions.

Applicable to multi-frequency quasi-periodic CMV matrices.

Abstract

We investigate the spectral structure of multi-frequency quasi-periodic CMV matrices with Verblunsky coefficients defined by shifts on the $d$-dimensional torus. Under the positive Lyapunov exponent regime and standard Diophantine frequency conditions, we establish that the spectrum of these operators contains intervals on the unit circle.