On the density of certain spectral points for a class of <inline-formula><tex-math id="M1">\begin{document}$ C^{2} $\end{document}</tex-math></inline-formula> quasiperiodic Schrödinger cocycles

Fan Wu; Linlin Fu; Jiahao Xu

doi:10.3934/dcds.2021201

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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

Fan Wu ^1, , Linlin Fu ^2, and Jiahao Xu ^3,,

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].

Keywords: Schrödinger cocycle, Cantor spectrum, $ C^2 $ cos-type potential, quasiperiodic, the density of spectral point.

Mathematics Subject Classification: Primary: 37A30.

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. Avila, J. 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. Avila, J. 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. Avila, D. 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. Avila, J. 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. Avila, J. 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. Avila, D. 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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On the density of certain spectral points for a class of $ C^{2} $ quasiperiodic Schrödinger cocycles

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