Purely Singular Continuous Spectrum for Limit-Periodic CMV Operators with Applications to Quantum Walks

TL;DR

This paper demonstrates that generic limit-periodic CMV operators typically have zero-measure Cantor spectrum, introduces a Craig--Simon type theorem for their density of states, and explores applications to quantum walks.

Contribution

It establishes the zero-measure Cantor spectrum for generic limit-periodic CMV operators and proves the optimality of a Craig--Simon type theorem for their density of states.

Findings

Generic limit-periodic CMV operators have zero-measure Cantor spectrum.

A Craig--Simon type theorem is proven for the density of states measure.

Applications to quantum walks with limit-periodic coin arrangements are discussed.

Abstract

We show that a generic element of a space of limit-periodic CMV operators has zero-measure Cantor spectrum. We also prove a Craig--Simon type theorem for the density of states measure associated with a stochastic family of CMV matrices and use our construction from the first part to prove that the Craig--Simon result is optimal in general. We discuss applications of these results to a quantum walk model where the coins are arranged according to a limit-periodic sequence. The key ingredient in these results is a new formula which may be viewed as a relationship between the density of states measure of a CMV matrix and its Schur function.