斜移 CMV 矩阵的李雅普诺夫行为和动态局域化

数学物理学报 ›› 2025, Vol. 45 ›› Issue (2): 334-346.

斜移 CMV 矩阵的李雅普诺夫行为和动态局域化

林艳雪()

烟台大学数学与信息科学学院山东烟台 264000;中国海洋大学数学科学学院山东青岛 266100

收稿日期:2024-01-05 修回日期:2024-09-16 出版日期:2025-04-26 发布日期:2025-04-09
作者简介:林艳雪,E-mail:yxlmath@sohu.com
基金资助:
国家自然科学基金(11571327);国家自然科学基金(11971059)

Dynamical Localization for the CMV Matrices with Verblunsky Coeffcients Defined by the Skew-Shift

Yanxue Lin()

Scool of Mathematics and Information Sciences, Yantai University, Shandong Yantai 264000; Scool of Mathematical Sciences, Ocean University of China, Shandong Qingdao 266100

Received:2024-01-05 Revised:2024-09-16 Online:2025-04-26 Published:2025-04-09
Supported by:
NSFC(11571327);NSFC(11971059)

摘要/Abstract

摘要：

该文主要证明对几乎所有频率, 当李雅普诺夫指数为正时, 斜移定义的 Verblunsky 系数生成的 CMV 矩阵的李雅普诺夫行为和动态局域化.

关键词: 斜移 CMV 矩阵, 李雅普诺夫行为, 动态局域化

Abstract:

In this paper, we prove the Lyapunov behavior and dynamical localization for the quasi-periodic CMV matrices with most frequencies and Verblunsky coefficients defined by the skew-shift, in the regime of positive Lyapunov exponents.

Key words: CMV Matrices, Lyapunov behavior, dynamical localization

中图分类号:

O193

林艳雪. 斜移 CMV 矩阵的李雅普诺夫行为和动态局域化[J]. 数学物理学报, 2025, 45(2): 334-346.

Yanxue Lin. Dynamical Localization for the CMV Matrices with Verblunsky Coeffcients Defined by the Skew-Shift[J]. Acta mathematica scientia,Series A, 2025, 45(2): 334-346.

参考文献 31

[1]	Avila A, Bochi J, Damanik D. Cantor spectrum for Schrödinger operators with potentials arising from generalized skew-shifts. Duke Math J, 2009, 146(2): 253-280
[2]	Avila A, Jitomirskaya S. Almost localization and almost reducibility. J Eur Math Soc, 2009, 12(1): 93-131
[3]	Bochner S, Martin W T. Several Complex Variables. Princeton NJ: Princeton University Press, 1948
[4]	Bourgain J. Green's Function Estimates for Lattice Schrödinger Operators and Applications. Princeton NJ: Princeton University Press, 2005
[5]	Bourgain J, Goldstein M. On nonperturbative localization with quasi-periodic potential. Ann of Math, 2000, 152(3): 835-879
[6]	Bourgain J, Goldstein M, Schlag W. Anderson localization for Schrödinger operators on $\mathbb{Z}$ with potentials given by the skew-shift. Comm Math Phys, 2001, 220: 583-621
[7]	Bourgain J, Goldstein M, Schlag W. Anderson localization for Schrödinger operators on $\mathbb{Z}^2$ with quasi-periodic potential. Acta Math, 2002, 188: 41-86
[8]	Bourgain J, Schlag W. Anderson localization for Schrödinger operators on $\mathbb{Z}$ with strongly mixing potentials. Comm Math Phys, 2000, 215(1): 143-175
[9]	Bucaj V, Damanik D, Fillman J, et al. Localization for the one-dimensional Anderson model via positivity and large deviations for the Lyapunov exponent. Trans Amer Math Soc, 2019, 372(5): 3619-3667
[10]	Cantero M J, Moral L, Grünbaum F A, Velázquez L. Matrix-valued Szegö polynomials and quantum random walks. Comm Pure Appl Math, 2010, 63(4): 464-507
[11]	Cantero M J, Moral L, Velázquez L. Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle. Linear Algebra Appl, 2003, 362: 29-56
[12]	Cedzich C, Werner A H. Anderson localization for electric quantum walks and skew-shift CMV matrices. Comm Math Phys, 2021, 387(3): 1257-1279
[13]	Chulaevsky V A, Sinai Ya G. Anderson localization for the $1$ -D discrete Schrödinger operator with two-frequency potential. Comm Math Phys, 1989, 125: 91-112
[14]	Damanik D. Schrödinger operators with dynamically defined potentials. Ergodic Theory Dynam Systems, 2017, 37(6): 1681-1764
[15]	Damanik D, Fillman J, Lukic M, Yessen W. Characterizations of uniform hyperbolicity and spectra of CMV matrices. Discrete Contin Dyn Syst S, 2016, 9(4): 1009-1023
[16]	Davis E B, Simon B. Eigenvalue estimates for non-normal matrices and the zeros of random orthogonal polynomials on the unit circle. J Approximation Theory, 2006, 141(2): 189-213
[17]	Fillman J, Ong D C. Purely singular continuous spectrum for limit-periodic CMV operators with applications to quantum walks. J Funct Anal, 2017, 272(12): 5107-5143
[18]	Goldstein M, Schlag W. Hölder continuity of the integrated density of the sates for quasiperodic Schrödinger equations and averages of shifts of subharmonic functions. Ann of Math, 2001, 154(1): 155-203
[19]	Guo S, Piao D. Lyapunov behavior and dynamical localization for quasi-periodic CMV matrices. Linear Algebra Appl, 2020, 606: 68-89
[20]	Kingman J. Subadditive ergodic theory. Ann Probab, 1973, 1: 883-899
[21]	Klein S. Anderson localization for the discrete one-dimensional quasi-periodic Schrödinger operator with potential defined by a Gevrey-class function. J Funct Anal, 2005, 218(2): 255-292
[22]	Krüger H. Orthogonal polynomials on the unit circle with Verblunsky coefficients defined by the skew-shift. Int Math Res Not, 2013, 2013(18): 4135-4169
[23]	Krüger H. The spectrum of skew-shift Schrödinger operators contains intervals. J Funct Anal, 2012, 262(3): 773-810
[24]	Lagendijk A, Tiggelen B, Wiersma D. Fifty years of Anderson localization. Phys Today, 2009, 62(8): 24-29
[25]	Lin Y X, Piao D X, Guo S Z. Anderson localization for the quasi-periodic CMV matrices with Verblunsky coefficients defined by the skew-shift. J Funct Anal, 2023, 285(4): 109975
[26]	Simon B. Orthogonal Polynomials on the Unit Circle. Province RI: American Mathematical Society, 2005
[27]	Sinai Y G. Anderson localization for one-dimensional difference Schrödinger operator with quasiperiodic potential. J Stat Phys, 1987, 46(5/6): 861-909
[28]	Tao K. Non-perturbative positive Lyapunov exponent of Schrödinger equations and its applications to skew-shift mapping. J Differential Equations, 2019, 266(6): 3559-3579
[29]	Wang F, Damanik D. Anderson localization for quasi-periodic CMV matrices and quantum walks. J Funct Anal, 2019, 276(6): 1978-2006 doi: 10.1016/j.jfa.2018.10.016
[30]	Zhang Z. Uniform hyperbolicity and its relation with spectral analysis of 1D discrete Schrödinger operators. J Spectr Theory, 2020, 10(4): 1471-1517
[31]	Zhu X W. Localization for the random CMV matrices. J Approx Theory, 2024, 298: 106008

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