Purely Singular Continuous Spectrum for Sturmian CMV Matrices via Strengthened Gordon Lemmas

TL;DR

This paper extends the Gordon Lemma to CMV matrices, proving they have purely singular continuous spectrum when generated by Sturmian Verblunsky coefficients, with implications for spectral theory and operator analysis.

Contribution

It generalizes the Gordon Lemma for CMV matrices without parity restrictions, enabling classification of their spectral types in Sturmian cases.

Findings

CMV matrices with Sturmian Verblunsky coefficients have purely singular continuous spectrum.

Spectrum is supported on a zero-measure Cantor set.

Results apply to CMV matrices generated by rotation codings.

Abstract

The Gordon Lemma refers to a class of results in spectral theory which prove that strong local repetitions in the structure of an operator preclude the existence of eigenvalues for said operator. We expand on recent work of Ong and prove versions of the Gordon Lemma which are valid for CMV matrices and which do not restrict the parity of scales upon which repetitions occur. The key ingredient in our approach is a formula of Damanik-Fillman-Lukic-Yessen which relates two classes of transfer matrices for a given CMV operator. There are many examples to which our result can be applied. We apply our theorem to complete the classification of the spectral type of CMV matrices with Sturmian Verblunsky coefficients; we prove that such CMV matrices have purely singular continuous spectrum supported on a Cantor set of zero Lebesgue measure for all (irrational) frequencies and all phases. We also discuss applications to CMV matrices with Verblunsky coefficients generated by general codings of rotations.