Abstract
In this article, the non-self dual extended Harper’s model with a Liouville frequency is considered. It is shown that the corresponding integrated density of states is \(\frac{1}{2}\)-Hölder continuous. As an application, the homogeneity of the spectrum is proven.
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Avila A. The absolutely continuous spectrum of the almost Mathieu operator. arXiv, 2008, 0810.2965
Avila A, Jitomirskaya S. Almost localization and almost reducibility. J Eur Math Soc, 2010, 12: 93–131
Avila A, Jitomirskaya S. Holder continuity of absolutely continuous spectral measures for one-frequency Schrodinger operators. Comm Math Phys, 2011, 301(2): 563–581
Avila A, Jitomirskaya S, Marx C A. Spectral theory of extended Harper’s model and a question by Erdos and Szekeres. Invent Math, 2017, 210(1): 283–339
Bourgain J. Holder regularity of integrated density of states for the almost Mathieu operator in a pertur-bative regime. Lett Math Phys, 2000, 51(2): 83–118
Cai A, Chavaudret C, You J G, Zhou Q. Sharp Holder continuity of the Lyapunov exponent of finitely differentiable quasi-periodic cocycles. Math Z, 2019, 29(3/4): 931–958
Damanik D, Goldstein M, Lukic M. The spectrum of a Schrodinger operator with small quasi-periodic potential is homogeneous. J Spectral Theory, 2016, 6(2): 415–427
Delyon F, Souillard B. The rotation number for finite difference operators and its properties. Comm Math Phys, 1983, 89(3): 415–426
Fillman J, Lukic M. Spectral homogeneity of limit-periodic Schrodinger operators. J Spectr Theory, 2017, 7(2): 387–406
Damanik D, Goldstein M, Schlag W, Voda M. Homogeneity of the spectrum for quasi-perioidic Schrodinger operators. J Eur Math Soc, 2018, 20(12): 3073–3111
Goldstein M, Schlag W. Holder continuity of the integrated density of states for quasi-periodic Schrodinger equations and averages of shifts of subharmonic functions. Ann Math, 2001, 154(1): 155–203
Goldstein M, Schlag W, Voda M. On localization and the spectrum of multi-frequency quasi-periodic operators. arXiv, 2016, 1610.00380
Hadj Amor S. Holder continuity of the rotation number for quasi-periodic co-cycles in SL(2,R). Comm Math Phys, 2009, 287(2): 565–588
Han R. Absence of point spectrum for the self-dual extended Harper’s model. Int Math Res Not, 2018, (9): 2801–2809
Han R. Dry Ten Martini problem for the non-self-dual extended Harper’s model. Trans Amer Math Soc, 2018, 370(1): 197–217
Han R, Jitomirskaya S. Full measure reducibility and localization for quasiperiodic Jacobi operators: a topological criterion. Adv Math, 2017, 319: 224–250
Han R, Yang F, Zhang S W. Spectral dimension for β-almost periodic singular Jacobi operators and the extended Harper’s model. arXiv, 2018, 1804.04322
Han R, Zhang S W. Optimal large deviation estimates and Holder regularity of the Lyapunov exponents for quasi-periodic Schrodinger cocycles. arXiv, 2018, 1803.02035
Jitomirskaya S, Koslover D A, Schulteis M S. Localization for a family of one-dimensional quasiperiodic operators of magnetic origin. Ann Henri Poincare, 2005, 6(1): 103–124
Johnson R, Moser J. The rotation number for almost periodic potentials. Comm Math Phys, 1983, 90(2): 317–318
Leguil M, You J G, Zhao Z Y, Zhou Q. Asymptotics of spectral gaps of quasi-periodic Schrodinger operators. arXiv, 2017, 1712.04700
Liu W C, Shi Y F. Upper bounds on the spectral gaps of quasi-periodic Schrodinger operators with Liouville frequencies. To appear in J Spectr Theory
Liu W C, Yuan X P. Holder continuity of the spectral measures for one-dimensional Schrodinger operator in exponential regime. J Math Phys, 2015, 56(1): 012701–21
Shi Y F, Yuan X P. Exponential decay of the lengths of the spectral gaps for the extended Harper’s model with a Liouvillean frequency. To appear in J Dynam Differential Equations
Tao K. Strong Birkhoff ergodic theorem for subharmonic functions with irrational shift and its application to analytic quasi-periodic cocycles. arXiv, 2018, 1805.00431
Tao K, Voda M. Holder continuity of the integrated density of states for quasi-periodic Jacobi operators. J Spectr Theory, 2017, 7(2): 361–386
Thouless D J. Bandwidths for a quasiperiodic tight-binding model. Phys Rev B, 1983, 28(8): 4272–4276
You J G, Zhang S W. Holder continuity of the Lyapunov exponent for analytic quasiperiodic Schrodinger cocycle with weak Liouville frequency. Ergodic Theory Dynam Systems, 2014, 34(4): 1395–1408
Acknowledgements
The authors would like to thank Prof. Xiaoping Yuan for his helpful suggestions.
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The second author was supported by China Postdoctoral Science Foundation (2018M641050).
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Jian, W., Shi, Y. Sharp Hölder Continuity of the Integrated Density of States for Extended Harper’s Model with a Liouville Frequency. Acta Math Sci 39, 1240–1254 (2019). https://doi.org/10.1007/s10473-019-0504-z
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DOI: https://doi.org/10.1007/s10473-019-0504-z