Orthogonal Polynomials on the Unit Circle with quasiperiodic Verblunsky Coefficients have generic purely singular continuous spectrum

2013/01/16 by Darren C. Ong, Ong, Darren C.
Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Mathematical functions and polynomials #Numerical methods in inverse problems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math-ph #math.DS #math.MP #math.SP

paper · pdf · doi:10.48550/arxiv.1301.3810

5 pages

arxiv created 2013/01/16 · openalex publication_date 2013/01/16 · arxiv updated 2013/01/17 · openalex created_date 2022/08/06 · openalex updated_date 2026/07/28

Abstract

As an application of the Gordon lemma for orthogonal polynomials on the unit circle, we prove that for a generic set of quasiperiodic Verblunsky coefficients the corresponding two-sided CMV operator has purely singular continuous spectrum. We use a similar argument to that of the Boshernitzan-Damanik result that establishes the corresponding theorem for the discrete Schrödinger operator.

Related

• A Borg-Type Theorem Associated with Orthogonal Polynomials on the Unit Circle
• Limit-Periodic Verblunsky Coefficients for Orthogonal Polynomials on the Unit Circle
• Aizenman's Theorem for Orthogonal Polynomials on the Unit Circle
• Orthogonal polynomials on the unit circle, q-Gamma weights, and discrete Painlevé equations
• Absolutely Continuous Spectrum for CMV Matrices With Small Quasi-Periodic Verblunsky Coefficients