Journal of Dynamics and Differential Equations

, Volume 31, Issue 4, pp 1921–1953 | Cite as

Exponential Decay of the Lengths of the Spectral Gaps for the Extended Harper’s Model with a Liouvillean Frequency

  • Yunfeng ShiEmail author
  • Xiaoping Yuan


In this paper, we study the non-self dual extended Harper’s model with a Liouvillean frequency. By establishing quantitative reducibility results together with the averaging method, we prove that the lengths of the spectral gaps decay exponentially.


Extended Harper’s model Liouvillean frequency Spectral gaps Quantitative reducibility 


  1. 1.
    Avila, A.: The absolutely continuous spectrum of the almost Mathieu operator. arXiv:0810.2965 (2008)
  2. 2.
    Avila, A., Jitomirskaya, S.: The Ten Martini problem. Ann. Math 170(1), 303–342 (2009)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Avila, A., Jitomirskaya, S.: Almost localization and almost reducibility. J. Eur. Math. Soc. 12, 93–131 (2010)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Avila, A., Jitomirskaya, S., Marx, C.A.: Spectral theory of extended Harper’s model and a question by Erdos and Szekeres. Invent. Math. 210(1), 283–339 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Avila, A., You, J., Zhou, Q.: Dry Ten Martini problem in non-critical case. PreprintGoogle Scholar
  6. 6.
    Béllissard, J., Simon, B.: Cantor spectrum for the almost Mathieu equation. J. Funct. Anal. 48(3), 408–419 (1982)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Choi, M.D., Elliott, G.A., Yui, N.: Gauss polynomials and the rotation algebra. Invent. Math. 99, 225–246 (1990)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Damanik, D., Goldstein, M.: On the inverse spectral problem for the quasi-periodic Schrödinger equation. Publ. Math. Inst. Hautes Études Sci. 119(1), 217–401 (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Damanik, D., Goldstein, M., Lukic, M.: The isospectral torus of quasi-periodic Schrödinger operators via periodic approximations. Invent. Math. 207(2), 895–980 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Delyon, F., Souillard, B.: The rotation number for finite difference operators and its properties. Commun. Math. Phys. 89(3), 415–426 (1983)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Eliasson, L.: Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation. Commun. Math. Phys. 146(3), 447–482 (1992)CrossRefGoogle Scholar
  12. 12.
    Hadj Amor, S.: Hölder continuity of the rotation number for quasi-periodic co-cycles in \(\text{ SL }(2,{\mathbb{R}})\). Commun. Math. Phys. 287(2), 565–588 (2009)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Han, R.: Dry Ten Martini problem for the non-self-dual extended Harper’s model. Trans. Am. Math. Soc. 370(1), 197–217 (2018)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Helffer, B., Sjöstrand, J.: Semiclassical analysis for Harper’s equation. III. Cantor structure of the spectrum. Mém. Soc. Math. France (N.S.) 39, 1–124 (1989)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Herman, M.-R.: Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d’un théorème d’Arnol’ d et de Moser sur le tore de dimension \(2\). Comment. Math. Helv. 58(3), 453–502 (1983)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Jian, W., Shi, Y.: Hölder continuity of the integrated density of states for extended Harper’s model with Liouville frequency. arXiv:1708.02670 [math.SP] (2017)
  17. 17.
    Jitomirskaya, S., Koslover, D.A., Schulteis, M.S.: Localization for a family of one-dimensional quasiperiodic operators of magnetic origin. Ann. Henri Poincaré 6(1), 103–124 (2005)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Jitomirskaya, S., Marx, C.A.: Analytic quasi-perodic cocycles with singularities and the Lyapunov exponent of extended Harper’s model. Commun. Math. Phys. 317(1), 237–267 (2012)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Johnson, R., Moser, J.: The rotation number for almost periodic potentials. Commun. Math. Phys. 90(2), 317–318 (1983)CrossRefGoogle Scholar
  20. 20.
    Krasovsky, I.: Central spectral gaps of the almost Mathieu operator. Commun. Math. Phys. 351(1), 419–439 (2017)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Leguil, M., You, J., Zhao, Z., Zhou, Q.: Asymptotics of spectral gaps of quasi-periodic Schrödinger operators. arXiv:1712.04700 [math.DS] (2017)
  22. 22.
    Liu, W., Shi, Y.: Upper bounds on the spectral gaps of quasi-periodic Schrödinger operators with Liouville frequencies. arXiv:1708.01760 [math.SP] (2017)
  23. 23.
    Liu, W., Yuan, X.: Spectral gaps of almost Mathieu operators in the exponential regime. J. Fractal Geom. 2(1), 1–51 (2015)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Parnovski, L., Shterenberg, R.: Perturbation theory for almost-periodic potentials I. One-dimensional case. arXiv:1711.03950 [math-ph] (2017)
  25. 25.
    Puig, J.: Cantor spectrum for the almost Mathieu operator. Commun. Math. Phys. 244(2), 297–309 (2004)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Puig, J.: A nonperturbative Eliasson’s reducibility theorem. Nonlinearity 19, 355–376 (2006)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Thouless, D.J.: Bandwidths for a quasiperiodic tight-binding model. Phys. Rev. B 28(8), 4272–4276 (1983)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Trent, T.T.: A new estimate for the vector valued corona problem. J. Funct. Anal. 189(1), 267–282 (2002)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Uchiyama, A.: Corona theorems for countably many functions and estimates for their solutions. Preprint (1990)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesFudan UniversityShanghaiPeople’s Republic of China

Personalised recommendations