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

TL;DR

This work analyzes multi-frequency quasi-periodic CMV matrices with Verblunsky coefficients generated by a $d$-torus shift under Diophantine frequencies and a positive Lyapunov exponent. The authors combine large deviation estimates, Lyapunov exponent control, Cartan-type theorems, Poisson formulas, finite-scale localization, and semialgebraic-set methods within a multiscale inductive framework to prove that the spectrum on the unit circle contains intervals. This CMV analogue of interval-spectrum results in MF-QP Schrödinger operators highlights the distinctive spectral structure arising from the unitary CMV setting and the complex analytic nature of Verblunsky coefficients. The results advance the understanding of spectral intervals in quasi-periodic CMV models and have implications for spectral theory of OPUC, and related quantum-walk dynamics.

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.

Theorems & Definitions (50)

• Theorem 1.1
• Remark 1.2
• Theorem 3.1
• Theorem 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5
• Lemma 3.6
• Remark 3.7
• Lemma 3.8