Spectral Approximation for Ergodic CMV Operators with an Application to Quantum Walks

TL;DR

This paper develops criteria for the spectral type of ergodic and limit-periodic CMV matrices, using variational estimates and inverse spectral theory, with applications to quantum walks.

Contribution

It introduces new criteria and variational estimates for the spectral properties of ergodic CMV matrices, extending results known for Schrödinger operators.

Findings

Criteria for fully supported absolutely continuous spectrum

Criteria for purely absolutely continuous spectrum in limit-periodic CMV matrices

Connection to quantum walks applications

Abstract

We establish concrete criteria for fully supported absolutely continuous spectrum for ergodic CMV matrices and purely absolutely continuous spectrum for limit-periodic CMV matrices. We proceed by proving several variational estimates on the measure of the spectrum and the vanishing set of the Lyapunov exponent for CMV matrices, which represent CMV analogues of results obtained for Schr\"odinger operators due to Y.\ Last in the early 1990s. Having done so, we combine those estimates with results from inverse spectral theory to obtain purely absolutely continuous spectrum.