May  2022, 42(5): 2525-2540. doi: 10.3934/dcds.2021201

On the density of certain spectral points for a class of $ C^{2} $ quasiperiodic Schrödinger cocycles

1. 

Department of Mathematics, Nanjing University, Nanjing 210093, China

2. 

Department of Mathematics, Sun Yat-sen University, Guangzhou 510275, China

3. 

Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300017, China

*Corresponding author: Jiahao Xu, 178881559@qq.com

Received  May 2021 Revised  November 2021 Published  May 2022 Early access  January 2022

Fund Project: The first author is supported by NNSF of China (Grants 11771205). The second author was supported by the Fundamental Research Funds for the Central Universities, Sun Yat-sen University (Grants 2021qntd21)

For $ C^2 $ cos-type potentials, large coupling constants, and fixed $ Diophantine $ frequency, we show that the density of the spectral points associated with the Schrödinger operator is larger than 0. In other words, for every fixed spectral point $ E $, $ \liminf\limits_{\epsilon\to 0}\frac{|(E-\epsilon,E+\epsilon)\bigcap\Sigma_{\alpha,\lambda\upsilon}|}{2\epsilon} = \beta $, where $ \beta\in [\frac{1}{2},1] $. Our approach is a further improvement on the papers [15] and [17].

Citation: Fan Wu, Linlin Fu, Jiahao Xu. On the density of certain spectral points for a class of $ C^{2} $ quasiperiodic Schrödinger cocycles. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2525-2540. doi: 10.3934/dcds.2021201
References:
[1]

A. AvilaJ. Bochi and D. Damanik, Cantor spectrum for Schrödinger operators with potentials arising from generalized skew-shifts, Duke Math. J., 146 (2009), 253-280.  doi: 10.1215/00127094-2008-065.

[2]

A. AvilaJ. Bochi and D. Damanik, Opening gaps in the spectrum of strictly ergodic schrödinger operators, J. Eur. Math. Soc., 14 (2012), 61-106.  doi: 10.4171/JEMS/296.

[3]

A. AvilaD. Damanik and Z. Zhang, Singular density of states measure for subshift and quasi-periodic Schrödinger operators, Comm. Math. Phys., 330 (2014), 469-498.  doi: 10.1007/s00220-014-1968-2.

[4]

A. Avila and S. Jitomirskaya, The ten martini problem, Ann. of Math., 170 (2009), 303-342.  doi: 10.4007/annals.2009.170.303.

[5]

A. Avila, J. You and Q. Zhou, Dry ten Martini problem in the non-critical case, preprint.

[6]

D. Damanik and D. Lenz, Zero-measure Cantor spectrum for Schrödinger operators with low-complexity potentials, J. Math. Pures Appl., 85 (2006), 671-686.  doi: 10.1016/j.matpur.2005.11.002.

[7]

D. Damanik and D. Lenz, A condition of boshernitzan and uniform convergence in the multiplicative ergodic theorem, Duke Math. J., 133 (2006), 95-123.  doi: 10.1215/S0012-7094-06-13314-8.

[8]

L. H. Eliasson, Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation, Comm. Math. Phys., 146 (1992), 447-482.  doi: 10.1007/BF02097013.

[9]

L. Ge, J. Y and X. Zhao, Arithmetic version of Anderson localization for quasiperiodic Schrödinger Operators with even cosine type potentials, arXiv: 2107.08547, 2021

[10]

M. Goldstein and W. Schlag, On resonances and the formation of gaps in the spectrum of quaiperiodic Schrödinger equations, Ann. of Math., 173 (2011), 337-475.  doi: 10.4007/annals.2011.173.1.9.

[11]

R. Johnson, Exponential dichotomy, rotation number, and linear differential operators with bounded coefficients, J. Differential Equations, 61 (1986), 54-78.  doi: 10.1016/0022-0396(86)90125-7.

[12]

R. Johnson and J. Moser, The rotation number for almost periodic potentials, Comm. Math. Phys., 90 (1983), 317-318.  doi: 10.1007/BF01205510.

[13]

B. Simon, Almost periodic Schrödinger operators: A review, Adv. in Appl. Math., 3 (1982), 463-490.  doi: 10.1016/S0196-8858(82)80018-3.

[14]

Y. G. Sinai, Anderson localization for one-dimensional difference Schrödinger operator with quasiperiodic potential, J. Statist. Phys., 46 (1987), 861-909.  doi: 10.1007/BF01011146.

[15]

J. Xu, L. Ge and Y. Wang, The Hölder continuity of Lyapunov exponents for a class of cos-type quasiperiodic Schrödinger cocycles, arXiv: 2006.03381v1, 2020.

[16]

Y. Wang and Z. Zhang, Uniform positivity and continuity of lyapunov exponents for a class of $C^2$ quasiperiodic Schrödinger cocycles, J. Funct. Anal., 268 (2015), 2525-2585.  doi: 10.1016/j.jfa.2015.01.003.

[17]

Y. Wang and Z. Zhang, Cantor spectrum for a class of $C^2$ quasiperiodic Schrödinger operators, Int. Math. Res. Not., 2017 (2017), 2300-2336.  doi: 10.1093/imrn/rnw079.

[18]

L. S. Young, Lyapunov exponents for some quasi-periodic cocycles, Ergodic Theory Dynam. Systems, 17 (1997), 483-504.  doi: 10.1017/S0143385797079170.

[19]

Z. Zhang, Uniform hyperbolicity and its relation with spectral analysis of 1D discrete Schrdinger operators, J. Spectr. Theory, 10 (2020), 1471-1517.  doi: 10.4171/JST/333.

show all references

References:
[1]

A. AvilaJ. Bochi and D. Damanik, Cantor spectrum for Schrödinger operators with potentials arising from generalized skew-shifts, Duke Math. J., 146 (2009), 253-280.  doi: 10.1215/00127094-2008-065.

[2]

A. AvilaJ. Bochi and D. Damanik, Opening gaps in the spectrum of strictly ergodic schrödinger operators, J. Eur. Math. Soc., 14 (2012), 61-106.  doi: 10.4171/JEMS/296.

[3]

A. AvilaD. Damanik and Z. Zhang, Singular density of states measure for subshift and quasi-periodic Schrödinger operators, Comm. Math. Phys., 330 (2014), 469-498.  doi: 10.1007/s00220-014-1968-2.

[4]

A. Avila and S. Jitomirskaya, The ten martini problem, Ann. of Math., 170 (2009), 303-342.  doi: 10.4007/annals.2009.170.303.

[5]

A. Avila, J. You and Q. Zhou, Dry ten Martini problem in the non-critical case, preprint.

[6]

D. Damanik and D. Lenz, Zero-measure Cantor spectrum for Schrödinger operators with low-complexity potentials, J. Math. Pures Appl., 85 (2006), 671-686.  doi: 10.1016/j.matpur.2005.11.002.

[7]

D. Damanik and D. Lenz, A condition of boshernitzan and uniform convergence in the multiplicative ergodic theorem, Duke Math. J., 133 (2006), 95-123.  doi: 10.1215/S0012-7094-06-13314-8.

[8]

L. H. Eliasson, Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation, Comm. Math. Phys., 146 (1992), 447-482.  doi: 10.1007/BF02097013.

[9]

L. Ge, J. Y and X. Zhao, Arithmetic version of Anderson localization for quasiperiodic Schrödinger Operators with even cosine type potentials, arXiv: 2107.08547, 2021

[10]

M. Goldstein and W. Schlag, On resonances and the formation of gaps in the spectrum of quaiperiodic Schrödinger equations, Ann. of Math., 173 (2011), 337-475.  doi: 10.4007/annals.2011.173.1.9.

[11]

R. Johnson, Exponential dichotomy, rotation number, and linear differential operators with bounded coefficients, J. Differential Equations, 61 (1986), 54-78.  doi: 10.1016/0022-0396(86)90125-7.

[12]

R. Johnson and J. Moser, The rotation number for almost periodic potentials, Comm. Math. Phys., 90 (1983), 317-318.  doi: 10.1007/BF01205510.

[13]

B. Simon, Almost periodic Schrödinger operators: A review, Adv. in Appl. Math., 3 (1982), 463-490.  doi: 10.1016/S0196-8858(82)80018-3.

[14]

Y. G. Sinai, Anderson localization for one-dimensional difference Schrödinger operator with quasiperiodic potential, J. Statist. Phys., 46 (1987), 861-909.  doi: 10.1007/BF01011146.

[15]

J. Xu, L. Ge and Y. Wang, The Hölder continuity of Lyapunov exponents for a class of cos-type quasiperiodic Schrödinger cocycles, arXiv: 2006.03381v1, 2020.

[16]

Y. Wang and Z. Zhang, Uniform positivity and continuity of lyapunov exponents for a class of $C^2$ quasiperiodic Schrödinger cocycles, J. Funct. Anal., 268 (2015), 2525-2585.  doi: 10.1016/j.jfa.2015.01.003.

[17]

Y. Wang and Z. Zhang, Cantor spectrum for a class of $C^2$ quasiperiodic Schrödinger operators, Int. Math. Res. Not., 2017 (2017), 2300-2336.  doi: 10.1093/imrn/rnw079.

[18]

L. S. Young, Lyapunov exponents for some quasi-periodic cocycles, Ergodic Theory Dynam. Systems, 17 (1997), 483-504.  doi: 10.1017/S0143385797079170.

[19]

Z. Zhang, Uniform hyperbolicity and its relation with spectral analysis of 1D discrete Schrdinger operators, J. Spectr. Theory, 10 (2020), 1471-1517.  doi: 10.4171/JST/333.

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