Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-27T12:28:05.316Z Has data issue: false hasContentIssue false
  • English
  • Français

Cantor spectrum for CMV and Jacobi matrices with coefficients arising from generalized skew-shifts

Part of: Dynamical systems with hyperbolic behavior Topological dynamics Special classes of linear operators General mathematical topics and methods in quantum theory

Published online by Cambridge University Press:  08 April 2021

HYUNKYU JUN Open the ORCID record for HYUNKYU JUN [Opens in a new window]
HYUNKYU JUN*
Affiliation:
Department of Mathematics, Rice University, Houston, TX77005, USA
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider continuous cocycles arising from CMV and Jacobi matrices. Assuming that the Verblunsky and Jacobi coefficients arise from generalized skew-shifts, we prove that uniform hyperbolicity of the associated cocycles is $C^0$ -dense. This implies that the associated CMV and Jacobi matrices have a Cantor spectrum for a generic continuous sampling map.

Keywords

CMV matricesJacobi matricesspectracocycleuniform hyperbolicity

MSC classification

Primary: 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
Secondary: 47B36: Jacobi (tridiagonal) operators (matrices) and generalizations 47B80: Random operators 81Q10: Selfadjoint operator theory in quantum theory, including spectral analysis 37B55: Nonautonomous dynamical systems
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 6 , June 2022 , pp. 2009 - 2027
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Avila, A., Bochi, J. and Damanik, D.. Cantor spectrum for Schrödinger operators with potentials arising from generalized skew-shifts. Duke Math. J. 146(2) (2009), 253–280.10.1215/00127094-2008-065CrossRefGoogle Scholar
Bochi, J.. Cocycles of isometries and denseness of domination. Q. J. Math. 66(3) (2015), 773–798.10.1093/qmath/hav020CrossRefGoogle Scholar
Bourget, O., Howland, J. S. and Joye, A.. Spectral analysis of unitary band matrices. Comm. Math. Phys. 234(2) (2003), 191–227.10.1007/s00220-002-0751-yCrossRefGoogle Scholar
Cantero, M. J., Moral, L. and Velázquez, L.. Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle. Linear Algebra Appl. 362 (2003), 29–56.10.1016/S0024-3795(02)00457-3CrossRefGoogle Scholar
Damanik, D.. Schrödinger operators with dynamically defined potentials. Ergod. Th. & Dynam. Sys. 37(6) (2017), 1681–1764.10.1017/etds.2015.120CrossRefGoogle Scholar
Damanik, D., Fillman, J., Lukic, M. and Yessen, W.. Characterizations of uniform hyperbolicity and spectra of CMV matrices. Discrete Contin. Dyn. Syst. Ser. S 9(4) (2016), 1009–1023.Google Scholar
Damanik, D. and Lenz, D.. Uniform Szegő cocycles over strictly ergodic subshifts. J. Approx. Theory 144(1) (2007), 133–138.10.1016/j.jat.2006.05.004CrossRefGoogle Scholar
Goldstein, M. and Schlag, W.. On resonances and the formation of gaps in the spectrum of quasi-periodic Schrödinger equations. Ann. of Math. (2) 173(1) (2011), 337–475.10.4007/annals.2011.173.1.9CrossRefGoogle Scholar
Goldstein, M., Schlag, W. and Voda, M.. On the spectrum of multi-frequency quasiperiodic Schrödinger operators with large coupling. Invent. Math. 217(2) (2019), 603–701.10.1007/s00222-019-00872-7CrossRefGoogle Scholar
Johnson, R. A.. Exponential dichotomy, rotation number, and linear differential operators with bounded coefficients. J. Differential Equations 61(1) (1986), 54–78.10.1016/0022-0396(86)90125-7CrossRefGoogle Scholar
Marx, C. A.. Dominated splittings and the spectrum of quasi-periodic Jacobi operators. Nonlinearity 27(12) (2014), 3059–3072.10.1088/0951-7715/27/12/3059CrossRefGoogle Scholar
Marx, C. A. and Jitomirskaya, S.. Dynamics and spectral theory of quasi-periodic Schrödinger-type operators. Ergod. Th. & Dynam. Sys. 37(8) (2017), 2353–2393.10.1017/etds.2016.16CrossRefGoogle Scholar
Simon, B.. Orthogonal Polynomials on the Unit Circle. Part 1. Classical Theory (American Mathematical Society Colloquium Publications, 54). American Mathematical Society, Providence, RI, 2005.Google Scholar
Simon, B.. Orthogonal Polynomials on the Unit Circle. Part 2. Spectral Theory (American Mathematical Society Colloquium Publications, 54). American Mathematical Society, Providence, RI, 2005.Google Scholar
Viana, M.. Lectures on Lyapunov Exponents (Cambridge Studies in Advanced Mathematics, 145). Cambridge University Press, Cambridge, 2014.10.1017/CBO9781139976602CrossRefGoogle Scholar
Wang, F. and Damanik, D.. Anderson localization for quasi-periodic CMV matrices and quantum walks. J. Funct. Anal. 276(6) (2019), 1978–2006.10.1016/j.jfa.2018.10.016CrossRefGoogle Scholar
Yoccoz, J.-C.. Some questions and remarks about $SL\left(2,\mathbb{R}\right)$ cocycles. Modern Dynamical Systems and Applications. Cambridge University Press, Cambridge, 2004, pp. 447–458.Google Scholar