Anderson localization for the multi-frequency quasi-periodic CMV matrices and quantum walks

TL;DR

This paper proves Anderson localization for multi-frequency quasi-periodic CMV matrices and quantum walks with analytic coefficients, overcoming resonance challenges through advanced mathematical techniques.

Contribution

It introduces a novel method combining semialgebraic sets, the Avalanche Principle, and Large Deviation Theorem to handle multi-frequency resonance issues.

Findings

Proves Anderson localization for multi-frequency quasi-periodic CMV matrices.

Establishes Anderson localization for related quantum walks.

Develops new techniques to eliminate double resonances in multi-frequency settings.

Abstract

In this paper we prove Anderson localization for multi-frequency quasi-periodic extended CMV matrices with analytic Verblunsky coefficients in the regime of positive Lyapunov exponents. By constructing a suitable semialgebraic set and combining the Avalanche Principle with a Large Deviation Theorem, we overcome the key obstruction of eliminating double resonances along the orbit, where multi-frequency potentials introduce significant challenges compared to the single-frequency case. As a direct application, we establish Anderson localization for corresponding analytic multi-frequency quasi-periodic quantum walks via unitary equivalence.