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Positive Hausdorff Dimensional Spectrum for the Critical Almost Mathieu Operator

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Abstract

We show that there exists a dense set of frequencies with positive Hausdorff dimension for which the Hausdorff dimension of the spectrum of the critical almost Mathieu operator is positive.

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References

  1. Avila A., Krikorian R.: Reducibility or non-uniform hyperbolicity for quasi-periodic Schrödinger cocycles. Ann. Math. 164, 911–940 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Avila A., Jitomirskaya S.: The Ten Martini Problem. Ann. Math. 170, 303–342 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Avila A., Last Y., Shamis M., Zhou Q.: On the abominable properties of the Almost Mathieu operator with well approximated frequencies, in preparation.

  5. Avila A., You J., Zhou Q.: Dry Ten Martini problem in the non-critical case, preprint.

  6. Avron J., van Mouche P., Simon B.: On the measure of the spectrum for the almost Mathieu operator. Commun. Math. Phys. 132, 103–118 (1990)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Avron J.E., Osadchy D., Seiler R.: A topological look at the quantum Hall effect. Commun. Math. Phys. 56, 38–42 (2003)

    Google Scholar 

  8. Bellissard J.: Le papillon de Hofstadter. Astérisque 206, 7–39 (1992)

    MATH  Google Scholar 

  9. Bell J., Stinchcombe R.B.: Hierarchical band clustering and fractal spectra in incommensurate systems. J. Phys. A 20, L739–L744 (1987)

    Article  MathSciNet  Google Scholar 

  10. Falconer K.: Techniques in Fractal Geometry. Wiley, New York (1997)

    MATH  Google Scholar 

  11. Geisel T., Ketzmerick R., Petshel G.: New class of level statistics in quantum systems with unbounded diffusion. Phys. Rev. Lett. 66, 1651–1654 (1991)

    Article  ADS  Google Scholar 

  12. Good I.J.: The fractional dimensional theory of continued fractions. Proc. Cambridge Philos. Soc. 37, 199–228 (1941)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Harper P.G.: Single band motion of conduction electrons in a uniform magnetic field. Proc. Phys. Soc. Lond. A 68, 874–892 (1955)

    Article  ADS  MATH  Google Scholar 

  14. Helffer B., Kerdelhué P.: On the total bandwidth for the rational Harper’s equation. Commun. Math. Phys. 173, 335–356 (1995)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Helffer B., Sjöstrand J.: Analyse semi-classique pour l’équation de Harper (avec application à l’équation de Schrödinger avec champ magnétique). Mém. Soc. Math. France 34, 1–113 (1988)

    MATH  Google Scholar 

  16. Helffer B., Sjöstrand J.: Analyse semi-classique pour l’équation de Harper II: Comportement semi-classique près d’un rationnel. Mém. Soc. Math. France 40, 1–139 (1990)

    MATH  Google Scholar 

  17. Helffer B., Sjöstrand J.: Semi-classical analysis for the Harper’s equation III: Cantor structure of the spectrum. Mém. Soc. Math. France 39, 1–124 (1989)

    MATH  Google Scholar 

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

  19. Johnson R., Moser J.: The rotation number for almost periodic potentials. Comm. Math. Phys. 84, 403–438 (1982)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Krasovsky I.: Central spectral gaps of the Almost Mathieu operator. Commun. Math. Phys. 351, 419–439 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Last Y.: Zero measure spectrum for the almost Mathieu Operator. Commun. Math. Phys. 164, 421–432 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Last Y., Shamis M: Zero Hausdorff dimension spectrum for the almost Mathieu operator. Commun. Math. Phys. 348, 729–750 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Osadchy D., Avron J. E.: Hofstadter butterfly as quantum phase diagram. J. Math. Phys. 42, 5665–5671 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Peierls R.: Zur Theorie des Diamagnetismus von Leitungselektronen. Z. Phys. 80, 763–791 (1933)

    Article  ADS  MATH  Google Scholar 

  25. Rauh A.: Degeneracy of Landau levels in crystals. Phys. Status Solidi B 65, 131–135 (1974)

    Article  ADS  Google Scholar 

  26. Royden H.L.: Real analysis. 3rd edn. Macmillan Publishing Company, New York (1988)

    MATH  Google Scholar 

  27. Tang C., Kohmoto M.: Global scaling properties of the spectrum for a quasiperiodic Schrödinger equation. Phys. Rev. B 34, 2041–2044 (1986)

    Article  ADS  Google Scholar 

  28. Thouless D.J., Kohmoto M., Nightingale M.P., Den Nijs M.: Quantized Hall conductance in a two dimensional periodic potential. Phys. Rev. Lett. 49, 405–408 (1982)

    Article  ADS  Google Scholar 

  29. Van Mouche P.: The coexistence problem for the discrete Mathieu operator. Commun. Math. Phys. 122(1), 23–33 (1989)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Wilkinson M., Austin E.J.: Spectral dimension and dynamics for Harper’s equation. Phys. Rev. B 50, 1420–1430 (1994)

    Article  ADS  Google Scholar 

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Acknowledgments

Q.-H. Liu was supported by NSFC grant (11571030). Y.-H. Qu was supported by NSFC Grant (11431007 and 11790273). Q. Zhou was partially supported by NSFC Grant (11671192), Specially appointed professor programme of Jiangsu province, and “Deng Feng Scholar Program B”of Nanjing University.

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Authors and Affiliations

  1. Laboratoire de Mathématiques Jean Leray, Université de Nantes and CNRS, 2 rue de la Houssinière, 44322, Nantes Cedex, France

    Bernard Helffer

  2. Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, Université Paris-Saclay, Orsay, France

    Bernard Helffer

  3. Department of Computer Science, Beijing Institute of Technology, Beijing, 100081, People’s Republic of China

    Qinghui Liu

  4. Department of Mathematics, Tsinghua University, Beijing, 100084, People’s Republic of China

    Yanhui Qu

  5. Department of Mathematics, Nanjing University, Nanjing, 210093, People’s Republic of China

    Qi Zhou

Authors
  1. Bernard Helffer
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  2. Qinghui Liu
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  3. Yanhui Qu
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  4. Qi Zhou
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Corresponding author

Correspondence to Yanhui Qu.

Additional information

Communicated by J. Marklof

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Helffer, B., Liu, Q., Qu, Y. et al. Positive Hausdorff Dimensional Spectrum for the Critical Almost Mathieu Operator. Commun. Math. Phys. 368, 369–382 (2019). https://doi.org/10.1007/s00220-018-3278-6

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  • DOI: https://doi.org/10.1007/s00220-018-3278-6

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