Singular Continuous Spectrum and Generic Full Spectral/Packing Dimension for Unbounded Quasiperiodic Schrödinger Operators


We proved that quasiperiodic Schrödinger operators with unbounded potentials \((H_{\alpha ,\theta }u)_n=u_{n+1}+u_{n-1}+ \frac{g(\theta +n\alpha )}{f(\theta +n\alpha )} u_n\) have purely singular continuous spectrum on the set \(\{E: 0<L(E)<\delta {(\alpha ,\theta ;f,g)}\}\), where \(\delta \) is an explicit function and L is the Lyapunov exponent. We assume that fg are Hölder continuous functions and f has finitely many zeros with weak non-degenerate assumptions. Moreover, we show that for generic \(\alpha \) and a.e. \(\theta \), the spectral measure of \(H_{\alpha ,\theta }\) has full spectral dimension and packing dimension.

This is a preview of subscription content, log in to check access.


  1. 1.

    We say \(\theta _0\in {{\mathbb {T}}}\) is a non-degenerate zero of \(f\in C^{k}({{\mathbb {T}}},{{\mathbb {C}}})\) if \(f(\theta _0)=0\) and \(f^{(k)}(\theta _0)\ne 0 \).

  2. 2.

    The sequence itself only depends on \(\beta (\alpha )\), while the largeness depends on \(\theta ,\alpha ,\beta ,\varepsilon ,\tau \).


  1. 1.

    Avila, A., Jitomirskaya, S.: The ten Martini problem. Ann. Math. 170(1), 303–342 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Avila, A., Jitomirskaya, S.: Almost localization and almost reducibility. J. Eur. Math. Soc. 12(1), 93–131 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Avila, A., You, J., Zhou, Q.: Sharp phase transitions for the almost Mathieu operator. Duke Math. J 166(14), 2697–2718 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Berry, M.: Incommensurability in an exactly-soluble quantal and classical model for a kicked rotator. Phys D Nonlinear Phenomena 10(3), 369–378 (1984)

    MathSciNet  Article  ADS  Google Scholar 

  5. 5.

    Bjerklv̈, K.: Dynamics of the quasiperiodic Schrodinger cocycle at the lowest energy in the spectrum. Commun. Math. Phys. 272(2), 397–442 (2007)

    Article  ADS  Google Scholar 

  6. 6.

    Bllissard, J., Lima, R., Scoppola, E.: Localization in v-dimensional incommensurate structures. Commun. Math. Phys. 88(4), 465–477 (1983)

    MathSciNet  Article  MATH  ADS  Google Scholar 

  7. 7.

    Boshernitzan, M., Damanik, D.: Generic continuous spectrum for ergodic Schrödinger operators. Commun. Math. Phys. 283(3), 647–662 (2008)

    Article  MATH  ADS  Google Scholar 

  8. 8.

    Cycon, H., Froese, R., Kirsch, W., Simon, B.: Schrödinger Operators with Application to Quantum Mechanics and Global Geometry, Texts and Monographs in Physics, Springer, Berlin (1987)

    Google Scholar 

  9. 9.

    Damanik, D.: Gordon-type arguments in the spectral theory of one-dimensional quasicrystals, Directions in Mathematical Quasicrystals, CRMMonogr, Ser., 13, Am. Math. Soc., Providence, RI, pp. 277–305, (2000)

  10. 10.

    Damanik, D.: A version of Gordon’s theorem for multi-dimensional Schrödinger operators. Trans. Am. Math. Soc. 356(2), 495–507 (2004)

    Article  MATH  Google Scholar 

  11. 11.

    del Rio, R., Jitomirskaya, S., Last, Y., Simon, B.: Operators with singular continuous spectrum, IV. Hausdorff dimensions, rank one perturbations, and localization. J. Anal. Math 69, 153–200 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Falconer, K.: Techniques in Fractal Geometry. Wiley, Chichester (1997)

    Google Scholar 

  13. 13.

    Furman, A.: On the multiplicative ergodic theorem for uniquely ergodic systems. Annal. l’Inst. Henri Poincare B Probab. Stat. 33(6), 797–815 (1997)

    MathSciNet  Article  MATH  ADS  Google Scholar 

  14. 14.

    Figotin, A.L., Pastur, L.A.: An exactly solvable model of a multidimensional incommensurate structure. Commun. Math. Phys. 95(4), 401–425 (1984)

    MathSciNet  Article  MATH  ADS  Google Scholar 

  15. 15.

    Ganeshan, S., Kechedzhi, K., Das Sarma, S.: Critical integer quantum hall topology and the integrable Maryland model as a topological quantum critical point. Phys. Rev. B 90(4), 041405 (2014)

    Article  ADS  Google Scholar 

  16. 16.

    Grempel, D., Fishman, S., Prange, R.: Localization in an incommensurate potential: an exactly solvable model. Phys. Rev. Lett. 49(11), 833 (1982)

    Article  ADS  Google Scholar 

  17. 17.

    Gordon, Y.A.: The point spectrum of the one-dimensional Schrödinger operator. Uspehi Mat. Nauk 31, 257–258 (1976)

    MathSciNet  Google Scholar 

  18. 18.

    Gordon, Y.A., Jaksic, V., Molchanov, S., Simon, B.: Spectral properties of random Schrodinger operators with unbounded potentials. Commun. Math. Phys. 157, 2350 (1993)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Han, R., Yang, F., Zhang, S.: Spectral Dimension for \(\beta \)-almost periodic singular Jacobi operators and the extended Harper’s model, to appear in Journal d’Analyse Mathématique (2019)

  20. 20.

    Janas, J., Naboko, S., Stolz, G.: Decay bounds on eigenfunctions and the singular spectrum of unbounded Jacobi matrices. Int. Math. Res. Not. 4, 736–764 (2009)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Jitomirskaya, S., Kachkovskiy, I.: All couplings localization for quasiperiodic operators with Lipschitz monotone potentials, to appear in J. Eur. Math. Soc. (2018)

  22. 22.

    Jitomirskaya, S., Last, Y.: Power-law subordinacy and singular spectra. I. Halfline Oper. Acta Math. 183, 171–189 (1999)

    Article  MATH  Google Scholar 

  23. 23.

    Jitomirskaya, S., Liu, W.: Arithmetic spectral transitions for the Maryland model. Commun. Pure Appl. Math. 70(6), 1025–1051 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Jitomirskaya, S., Mavi, R.: Dynamical bounds for quasiperiodic Schrödinger operators with rough potentials. Int. Math. Res. Not. 1, 96–120 (2017)

    MATH  Google Scholar 

  25. 25.

    Jitomirskaya, S., Yang, F.: Singular continuous spectrum for singular potentials. Commun. Math. Phys. 351(3), 1127–1135 (2017)

    MathSciNet  Article  MATH  ADS  Google Scholar 

  26. 26.

    Jitomirskaya, S., Zhang, S.: Quantitative continuity of singular continuous spectral measures and arithmetic criteria for quasiperiodic Schrödinger operators, preprint. arXiv:1510.07086 (2015)

  27. 27.

    Khinchin, A.Y.: Continued Fractions. University of Chicago, Chicago (1964)

    Google Scholar 

  28. 28.

    Kirsch, W., Molchanov, S.A., Pastur, L.A.: The one-dimensional Schrodinger operator with unbounded potential: the pure point spectrum. Funct. Anal. Appl. 24, 176–86 (1990)

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Kirsch, W., Molchanov, S.A., Pastur, L.A.: One-dimensional Schrödinger operators with high potential barriers. Oper. Theory Adv. Appl. 57, 163–70 (1992)

    MATH  Google Scholar 

  30. 30.

    Klein, S.: Anderson localization for the discrete one-dimensional quasi-periodic Schrodinger operator with potential defined by a Gevrey-class function. J. Funct. Anal. 218(2), 255–292 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  31. 31.

    Simon, B., Spencer, T.: Trace class perturbations and the absence of absolutely continuous spectra. Commun. Math. Phys. 125(1), 113–125 (1989)

    MathSciNet  Article  MATH  ADS  Google Scholar 

  32. 32.

    Simon, B.: Almost periodic Schrödinger operators. IV. The Maryland model. Ann. Phys. 159(1), 157–183 (1985)

    Article  MATH  ADS  Google Scholar 

  33. 33.

    Simon, B.: Equilibrium measures and capacities in spectral theory. Inv. Probl. Imaging 1, 376–382 (2007)

    MathSciNet  Google Scholar 

  34. 34.

    Wang, Y., Zhang, Z.: Cantor spectrum for a class of \(C^2\) quasiperiodic Schrödinger operators. Int. Math. Res. Not. 8, 2300–2336 (2017)

    MATH  Google Scholar 

Download references


The authors would like to thank Rui Han for useful discussions. The authors would also like to thank Ilya Kachkovskiy for mentioning important studies of unbounded Schrödinger operators to us. Last but not least, the authors would like to thank Svetlana Jitomirskaya for reading the early manuscript and useful comments. F. Y. would like to thank the Institute for Advanced Study, Princeton, for its hospitality during the 2017–2018 academic year. F. Y. was supported in part by NSF grant DMS-1638352. S. Z. was supported in part by NSF Grant DMS-1600065 and DMS-1758326 and by a postdoctoral fellowship from the MSU Institute for Mathematical and Theoretical Physics.

Author information



Corresponding author

Correspondence to Shiwen Zhang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by Jean Bellissard.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Yang, F., Zhang, S. Singular Continuous Spectrum and Generic Full Spectral/Packing Dimension for Unbounded Quasiperiodic Schrödinger Operators. Ann. Henri Poincaré 20, 2481–2494 (2019).

Download citation