Annales Henri Poincaré

, Volume 20, Issue 11, pp 3563–3601 | Cite as

Construction of QuasiPeriodic Schrödinger Operators with Cantor Spectrum

  • Xuanji Hou
  • Yuan Shan
  • Jiangong YouEmail author


It is well known that, for fixed Diophantine frequencies and generic small smooth or analytic quasiperiodic potentials, both continuous and discrete Schrödinger operators have Cantor spectrum. Although there have been several examples of Schrödinger operators with Cantor spectrum since Moser’s pioneering work (Bellissard, in: Luck, Moussa, Waldschmidt (eds) Number theory and physics (Les Houches, 1989), Springer, Berlin, 1990; Bellissard et al. in Phys Rev Lett 49:701–704, 1982; Damanik et al. in Ann Henri Poincaré 15:1123–1144, 2014, J Spectr Theory 7:1101–1118, 2017; Moser in Comment Math Helv 56:198–224, 1981); however, so far there is no concrete quasiperiodic example in the continuous case, and there is no concrete quasiperiodic example in the discrete case besides the cosine-like potentials (Avila and Jitomirskaya in Ann Math 170:303–342, 2009; Puig in Commun Math Phys 244:297–309, 2004; Sinai in J Stat Phys 46:861–909; 1987; Wang and Zhang in Int Math Res Not 2017:2300–2336, 2017). In this paper, we present a strategy for explicitly constructing quasiperiodic Schrödinger operators with Cantor spectrum.



X. Hou was partially supported by NNSF of China (Grant 11371019, 11671395) and Self-Determined Research Funds of Central China Normal University (CCNU19QN078). Y. Shan was partially supported by NNSF of China (Grant 11701285) and Natural Science Foundation of Jiangsu Province, China (Grant BK20161053). J. You was partially supported by NNSF of China (11871286) and Nankai Zhide Foundation.


  1. 1.
    Avila, A.: On the spectrum and Lyapunov exponent of limit periodic Schrödinger operators. Commun. Math. Phys. 288, 907–918 (2009)ADSzbMATHGoogle Scholar
  2. 2.
    Avila, A., Bochi, J., Damanik, D.: Cantor spectrum for Schrödinger operators with potentials arising from generalized skew-shifts. Duke Math. J. 146, 253–280 (2009)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Avila, A., Krikorian, R.: Reducibility or nonuniform hyperbolicity for quasiperiodic Schrödinger cocycles. Ann. Math. 164, 911–940 (2006)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Avila, A., Jitomirskaya, S.: The ten martini problem. Ann. Math. 170, 303–342 (2009)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Avron, J., Simon, B.: Almost periodic Schrödinger operators. I. Limit periodic potentials. Commun. Math. Phys. 82, 101–120 (1981)ADSzbMATHGoogle Scholar
  6. 6.
    Amor, S.H.: Hölder Continuity of the rotation number for quasi-periodic co-cycles in \(SL(2,{\mathbb{R}})\). Commun. Math. Phys. 287, 565–588 (2009)ADSzbMATHGoogle Scholar
  7. 7.
    Bellissard, J.: Spectral properties of Schrödinger’s operator with a Thue-Morse potential. In: Luck, J.M., Moussa, P., Waldschmidt, M. (eds.) Number theory and physics (Les Houches, 1989). Springer proc. phys., vol. 47, pp. 140–150. Springer, Berlin (1990)Google Scholar
  8. 8.
    Bellissard, J.: Gap labeling theorems for Schrödinger’s operators. In: Luck, J.M., Moussa, P., Waldschmidt, M. (eds.) From number theory to physics, Les Houches March, vol. 89, pp. 538–630. Springer, Berlin (1993)Google Scholar
  9. 9.
    Bellissard, J., Bovier, A., Ghez, J.M.: Spectral properties of a tight binding Hamiltonian with period doubling potential. Commun. Math. Phys. 135, 379–399 (1991)ADSMathSciNetzbMATHGoogle Scholar
  10. 10.
    Bellissard, J., Bessis, D., Moussa, P.: Chaotic states of almost periodic Schrödinger operators. Phys. Rev. Lett. 49, 701–704 (1982)ADSMathSciNetGoogle Scholar
  11. 11.
    Bovier, A., Ghez, J.M.: Spectral properties of one-dimensional Schrödinger operators with potentials generated by substitutions. Commun. Math. Phys. 158, 45–66; Erratum: Commun. Math. Phys. 166(1994), 431–432 (1993)Google Scholar
  12. 12.
    Bellissard, J., Iochum, B., Scoppola, E., Testard, D.: Spectral properties of one-dimensional quasicrystals. Commun. Math. Phys. 125, 527–543 (1989)ADSzbMATHGoogle Scholar
  13. 13.
    Cai, A., Chavaudret, C., You, J., Zhou, Q.: Sharp Hölder continuity of the Lyapunov exponent of finitely differentiable quasi-periodic cocycles. Math. Z. 291, 931–958 (2019)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Damanik, D.: Strictly ergodic subshifts and associated operators. In: Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday, Proc. Sympos. Pure Math. vol. 76, Part 2. Amer. Math. Soc., Providence, pp. 505–538 (2007)Google Scholar
  15. 15.
    Damanik, D.: Schrödinger operators with dynamically defined potentials: a survey. Ergod. Theory Dyn. Syst. 37, 1–84 (2014)Google Scholar
  16. 16.
    Damanik, D., Fillman, J., Gorodetski, A.: Continuum Schrödinger operators associated with aperiodic subshifts. Ann. Henri Poincaré 15, 1123–1144 (2014)ADSMathSciNetzbMATHGoogle Scholar
  17. 17.
    Damanik, D., Fillman, J., Lukic, M.: Limit-periodic continuum Schrödinger operators with zero measure cantor spectrum. J. Spectr. Theory 7, 1101–1118 (2017)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Damanik, D., Gan, Z.: Spectral properties of limit-periodic Schrödinger operators. Commun. Pure Appl. Anal. 10, 859–871 (2011)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Damanik, D., Goldstein, M.: On the inverse spectral problem for the quasi-periodic Schrödinger equation. Publ. Math. Inst. Hautes Études Sci. 119, 217–401 (2014)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Damanik, D., Goldstein, M., Lukic, M.: The isospectral torus of quasi-periodic Schrödinger operators via periodic approximations. Invent. Math. 207, 895–980 (2017)ADSMathSciNetzbMATHGoogle Scholar
  21. 21.
    Damanik, D., Lenz, D.: A condition of Boshernitzan and uniform convergence in the multiplicative ergodic theorem. Duke Math. J. 133, 95–123 (2006)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Damanik, D., Lenz, D.: Zero-measure Cantor spectrum for Schrödinger operators with low complexity potentials. J. Math. Pures Appl. 85, 671–686 (2006)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Eliasson, H.: Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation. Commun. Math. Phys. 146, 447–482 (1992)ADSzbMATHGoogle Scholar
  24. 24.
    Fayad, B., Krikorian, R.: Rigidity results for quasiperiodic \(SL(2,{\mathbb{R}})-\) cocyles. J. Mod. Dyn. 3, 479–510 (2009)zbMATHGoogle Scholar
  25. 25.
    Fillman, J., Lukic, M.: Spectral homogeneity of limit-periodic Schrödinger operators. J. Spectr. Theory 7, 387–406 (2017)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Goldstein, M., Schlag, W.: On resonances and the formation of gaps in the spectrum of quasi-periodic Schrödinger equations. Ann. Math. 173, 337–475 (2011)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Goldstein, M., Schlag, W., Voda, M.: On the spectrum of multi-frequency quasiperiodic Schrödinger operators with large coupling. arXiv:1708.09711
  28. 28.
    Hou, X., Popov, G.: Rigidity of reducibility of Gevrey quasi-periodic cocycles on \(U(n)\). Bull. Soc. Math. France 144, 1–52 (2016)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Hou, X., You, J.: Almost reducibility and non-perturbative reducibility of quasi-periodic linear systems. Invent. Math. 190, 209–260 (2012)ADSMathSciNetzbMATHGoogle Scholar
  30. 30.
    Johnson, R., Moser, J.: The rotation number for almost periodic potentials. Commun. Math. Phys. 84, 403–438 (1982)ADSMathSciNetzbMATHGoogle Scholar
  31. 31.
    Karpeshina, Y.: Spectral properties of the periodic magnetic Schödinger operator in the high-energy region. Two-dimensional case. Commun. Math. Phys. 251, 473–514 (2004)ADSMathSciNetzbMATHGoogle Scholar
  32. 32.
    Kunz, H., Livi, R., Sütő, A.: Cantor spectrum and singular continuity for a hierarchical Hamiltonian. Commun. Math. Phys. 122, 643–679 (1989)ADSMathSciNetzbMATHGoogle Scholar
  33. 33.
    Lenz, D.: Singular continuous spectrum of Lebesgue measure zero for one-dimensional quasicrystals. Commun. Math. Phys. 227, 119–130 (2002)ADSzbMATHGoogle Scholar
  34. 34.
    Liu, Q., Qu, Y.: Uniform convergence of Schrödinger cocycles over simple Toeplitz subshift. Ann. Henri Poincaré 12, 153–172 (2011)ADSMathSciNetzbMATHGoogle Scholar
  35. 35.
    Liu, Q., Qu, Y.: Uniform convergence of Schrödinger cocycles over bounded Toeplitz subshift. Ann. Henri Poincaré 13, 1483–1500 (2012)ADSMathSciNetzbMATHGoogle Scholar
  36. 36.
    Liu, Q., Tan, B., Wen, Z., Wu, J.: Measure zero spectrum of a class of Schrödinger operators. J. Stat. Phys. 106, 681–691 (2002)zbMATHGoogle Scholar
  37. 37.
    Leguil, M., You, J., Zhao, Z., Zhou, Q.: Asymptotics of spectral gaps of quasiperiodic Schrödinger operators. arXiv: 1712.04700
  38. 38.
    Moser, J.: An example of a Schrödinger equation with an almost periodic potential and nowhere dense spectrum. Comment. Math. Helv. 56, 198–224 (1981)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Moser, J., Pöschel, J.: An extension of a result by Dinaburg and Sinai on quasiperiodic potentials. Comment. Math. Helv. 59, 39–85 (1984)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Molchanov, S., Chulaevsky, V.: The structure of a spectrum of the lacunary-limit-periodic Schrödinger operator. Funct. Anal. Appl. 18, 343–344 (1984)Google Scholar
  41. 41.
    Puig, J.: Cantor spectrum for the almost Mathieu operator. Commun. Math. Phys. 244, 297–309 (2004)ADSMathSciNetzbMATHGoogle Scholar
  42. 42.
    Puig, J.: A nonperturbative Eliasson’s reducibility theorem. Nonlinearity 19, 355–376 (2006)ADSMathSciNetzbMATHGoogle Scholar
  43. 43.
    Pastur, L., Tkachenko, V.: Spectral theory of a class of one-dimensional Schrödinger operators with limit-periodic potentials. Trudy Moskov. Mat. Obshch. 51, 114–168 (1988)Google Scholar
  44. 44.
    Simon, B.: Kotani theory for one-dimensional stochastic Jacobi matrices. Commun. Math. Phys. 89, 227–234 (1983)ADSMathSciNetzbMATHGoogle Scholar
  45. 45.
    Sinai, Y.G.: Anderson localization for one-dimensional difference Schrödinger operator with quasiperiodic potential. J. Stat. Phys. 46, 861–909 (1987)ADSGoogle Scholar
  46. 46.
    Sire, C.: Electronic spectrum of a 2D quasi-crystal related to the octagonal quasi-periodic tiling. EPL (Europhys. Lett.) 10, 483–488 (1989)ADSGoogle Scholar
  47. 47.
    Skriganov, M.: Geometric and Arithmetic Methods in the Spectral Theory of Multidimensional Periodic Operators. American Mathematical Soc, Providence (1987)zbMATHGoogle Scholar
  48. 48.
    Sütő, A.: The spectrum of quasiperiodic Shrödinger operator. Commun. Math. Phys. 111, 409–415 (1987)ADSzbMATHGoogle Scholar
  49. 49.
    Sütő, A.: Singular continuous spectrum on a Cantor set of zero Lebesgue measure for the Fibonacci Hamiltonian. J. Stat. Phys. 56, 525–531 (1989)ADSMathSciNetzbMATHGoogle Scholar
  50. 50.
    Wang, Y., Zhang, Z.: Cantor spectrum for a class of \(C^2\) quasiperiodic Schrödinger operators. Int. Math. Res. Not. 2017, 2300–2336 (2017)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mathematics and Statistics Hubei Key Laboratory of Mathematical SciencesCentral China Normal UniversityWuhanPeople’s Republic of China
  2. 2.School of Statistics and MathematicsNanjing Audit UniversityNanjingPeople’s Republic of China
  3. 3.Chern Institute of Mathematics and LPMCNankai UniversityTianjinPeople’s Republic of China

Personalised recommendations