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On the spectrum of multi-frequency quasiperiodic Schrödinger operators with large coupling

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Abstract

We study multi-frequency quasiperiodic Schrödinger operators on \({\mathbb {Z}}\). We prove that for a large real analytic potential satisfying certain restrictions the spectrum consists of a single interval. The result is a consequence of a criterion for the spectrum to contain an interval at a given location that we establish non-perturbatively in the regime of positive Lyapunov exponent.

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References

  1. Avila, A., Jitomirskaya, S.: The ten martini problem. Ann. Math. (2) 170(1), 303–342 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Avila, A.: Global theory of one-frequency Schrödinger operators. Acta Math. 215(1), 1–54 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bourgain, J.: On the spectrum of lattice Schrödinger operators with deterministic potential. J. Anal. Math. 87, 37–75 (2002). Dedicated to the memory of Thomas H. Wolff

    Article  MathSciNet  MATH  Google Scholar 

  5. Bourgain, J.: Green’s Function Estimates for Lattice Schrödinger Operators and Applications. Annals of Mathematics Studies, vol. 158. Princeton University Press, Princeton, NJ (2005)

    Book  MATH  Google Scholar 

  6. Bourgain, J.: Positivity and continuity of the Lyapounov exponent for shifts on \(\mathbb{T}^d\) with arbitrary frequency vector and real analytic potential. J. Anal. Math. 96, 313–355 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bourgain, J.: Anderson localization for quasi-periodic lattice Schrödinger operators on \(\mathbb{Z}^d\), \(d\) arbitrary. Geom. Funct. Anal. 17(3), 682–706 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bourgain, J., Goldstein, M.: On nonperturbative localization with quasi-periodic potential. Ann. Math. (2) 152(3), 835–879 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chulaevsky, V.A., Sinaĭ, Y.G.: Anderson localization for the \(1\)-D discrete Schrödinger operator with two-frequency potential. Commun. Math. Phys. 125(1), 91–112 (1989)

    Article  MATH  Google Scholar 

  10. Duarte, P., Klein, S.: Continuity, positivity and simplicity of the Lyapunov exponents for quasi-periodic cocycles. ArXiv e-prints, March (2016)

  11. Dinaburg, E.I., Sinaĭ, J.G.: The one-dimensional Schrödinger equation with quasiperiodic potential. Funkcional. Anal. i Priložen. 9(4), 8–21 (1975)

    MathSciNet  Google Scholar 

  12. Fröhlich, J., Spencer, T., Wittwer, P.: Localization for a class of one-dimensional quasi-periodic Schrödinger operators. Commun. Math. Phys. 132(1), 5–25 (1990)

    Article  MATH  Google Scholar 

  13. Goldstein, M., Schlag, W.: Hölder continuity of the integrated density of states for quasi-periodic Schrödinger equations and averages of shifts of subharmonic functions. Ann. Math. (2) 154(1), 155–203 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Goldstein, M., Schlag, W.: Fine properties of the integrated density of states and a quantitative separation property of the Dirichlet eigenvalues. Geom. Funct. Anal. 18(3), 755–869 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Goldstein, M., Schlag, W.: On resonances and the formation of gaps in the spectrum of quasi-periodic Schrödinger equations. Ann. Math. (2) 173(1), 337–475 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Goldstein, M., Schlag, W., Voda, M.: On localization and spectrum of multi-frequency quasi-periodic operators. ArXiv e-prints (2016)

  17. Jitomirskaya, S., Marx, C.A.: Dynamics and spectral theory of quasi-periodic Schrödinger-type operators. Ergod. Theory Dyn. Syst. 37(8), 2353–2393 (2017)

    Article  MATH  Google Scholar 

  18. Krüger, H.: The spectrum of skew-shift Schrödinger operators contains intervals. J. Funct. Anal. 262(3), 773–810 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Karpeshina, Y., Shterenberg, R.: Extended states for the Schrödinger operator with quasi-periodic potential in dimension two. ArXiv e-prints, August (2014)

  20. Lang, S.: Algebra. Graduate Texts in Mathematics, vol. 211, 3rd edn. Springer, New York (2002)

    Book  MATH  Google Scholar 

  21. Marden, M.: Geometry of Polynomials. Mathematical Surveys, No. 3, 2nd edn. American Mathematical Society, Providence, RI (1966)

    MATH  Google Scholar 

  22. Puig, J.: Cantor spectrum for the almost Mathieu operator. Commun. Math. Phys. 244(2), 297–309 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. Sinaĭ, Y.G.: Anderson localization for one-dimensional difference Schrödinger operator with quasiperiodic potential. J. Stat. Phys. 46(5–6), 861–909 (1987)

    Article  Google Scholar 

  24. Wang, Y., Zhang, Z.: Uniform positivity and continuity of Lyapunov exponents for a class of \(C^2\)-quasiperiodic Schrödinger cocycles. J. Funct. Anal. 268, 2525–2585 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Toronto, Toronto, ON, M5S 1A1, Canada

    Michael Goldstein & Mircea Voda

  2. Department of Mathematics, Yale University, 10 Hillhouse Ave., New Haven, CT, 06511, USA

    Wilhelm Schlag

  3. Department of Mathematics, The University of Chicago, 5734 S. University Ave., Chicago, IL, 60637, USA

    Mircea Voda

Authors
  1. Michael Goldstein
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  2. Wilhelm Schlag
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  3. Mircea Voda
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Correspondence to Wilhelm Schlag.

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The first author was partially supported by an NSERC Grant. The second author was partially supported by the NSF, DMS-1500696.

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Goldstein, M., Schlag, W. & Voda, M. On the spectrum of multi-frequency quasiperiodic Schrödinger operators with large coupling. Invent. math. 217, 603–701 (2019). https://doi.org/10.1007/s00222-019-00872-7

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  • DOI: https://doi.org/10.1007/s00222-019-00872-7

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