Skip to main content
SpringerLink
Log in

Multidimensional Almost-Periodic Schrödinger Operators with Cantor Spectrum

  • Published:
Annales Henri Poincaré Aims and scope Submit manuscript

Abstract

We construct multidimensional almost-periodic Schrödinger operators whose spectrum has zero lower box-counting dimension. In particular, the spectrum in these cases is a generalized Cantor set of zero Lebesgue measure.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Avila, A.: On the spectrum and Lyapunov exponent of limit-periodic Schrödinger operators. Commun. Math. Phys. 288, 907–918 (2009)

    Article  ADS  MATH  Google Scholar 

  2. Avron, J., Simon, B.: Almost periodic Schrödinger operators. I. Limit periodic potentials. Commun. Math. Phys. 82, 101–120 (1981)

    Article  ADS  MATH  Google Scholar 

  3. Damanik, D., Fillman, J., Gorodetski, A.: Continuum Schrödinger operators associated with aperiodic subshifts. Ann. Henri Poincaré 15, 1123–1144 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Damanik, D., Fillman, J., Lukic, M.: Limit-periodic continuum Schrödinger operators with zero-measure Cantor spectrum. J. Spectr. Theory 7, 1101–1118 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  5. Eliasson, L.H.: Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation. Commun. Math. Phys. 146, 447–482 (1992)

    Article  ADS  MATH  Google Scholar 

  6. Embree, M., Fillman, J.: Spectra of discrete two-dimensional periodic Schrödinger operators with small potentials. J. Spectr. Theory. (in press). arXiv:1701.00863

  7. Exner, P., Turek, O.: Periodic quantum graphs from the Bethe–Sommerfeld perspective. J. Phys. A Math. Theor. 50, 455201 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Fillman, J., Han, R.: Discrete Bethe–Sommerfeld conjecture for triangular, square, and hexagonal lattices. Preprint arXiv:1806.01988

  9. Fillman, J., Lukic, M.: Spectral homogeneity of limit-periodic Schrödinger operators. J. Spectr. Theory 7, 387–406 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gordon, A.: On the point spectrum of the one-dimensional Schrödinger operator. Usp. Math. Nauk 31, 257–258 (1976)

    Google Scholar 

  11. Hadj Amor, S.: Hölder continuity of the rotation number for quasi-periodic co-cycles in \(SL(2, \mathbb{R})\). Commun. Math. Phys. 287, 565–588 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Han, R., Jitomirskaya, S.: Discrete Bethe–Sommerfeld conjecture. Commun. Math. Phys. 361, 205–216 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Helffer, B., Mohamed, A.: Asymptotics of the density of states for the Schrödinger operator with periodic electric potential. Duke Math. J. 92, 1–60 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  14. Karpeshina, Y.: Perturbation Theory for the Schrödinger Operator with a Periodic Potential, Lecture Notes in Mathematics, vol. 1663. Springer, Berlin (1997)

    Book  MATH  Google Scholar 

  15. Karpeshina, Y., Lee, Y.-R.: Spectral properties of a limit-periodic Schrödinger operator in dimension two. J. Anal. Math. 120, 1–84 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Karpeshina, Y., Shterenberg, R.: Extended states for the Schrödinger operator with quasi-periodic potential in dimension two. To appear in Mem. Am. Math. Soc. arXiv:1408.5660

  17. Krüger, H.: Periodic and limit-periodic discrete Schrödinger operators. Preprint arXiv:1108.1584

  18. Kuchment, P.: An overview of periodic elliptic operators. Bull. Am. Math. Soc. 53, 343–414 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Molchanov, S., Chulaevsky, V.: The structure of a spectrum of the lacunary-limit-periodic Schrödinger operator. Funct. Anal. Appl. 18, 343–344 (1984)

    Article  Google Scholar 

  20. Moser, J.: An example of a Schrödinger equation with almost periodic potential and nowhere dense spectrum. Comment. Math. Helv. 56, 198–224 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  21. Parnovski, L.: Bethe–Sommerfeld conjecture. Ann. Henri Poincaré 9, 457–508 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Pastur, L., Tkachenko, V.A.: On the spectral theory of the one-dimensional Schrödinger operator with limit-periodic potential (Russian). Dokl. Akad. Nauk SSSR 279, 1050–1053 (1984)

    MathSciNet  Google Scholar 

  23. 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 

  24. Popov, V.N., Skriganov, M.: A remark on the spectral structure of the two dimensional Schrödinger operator with a periodic potential. Zap. Nauchn. Sem. LOMI AN SSSR 109, 131–133 (1981). (in Russian)

    MATH  Google Scholar 

  25. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press, New York (1978)

    MATH  Google Scholar 

  26. Simon, B.: On the genericity of nonvanishing instability intervals in Hill’s equation. Ann. Inst. H. Poincaré Sect. A (N.S.) 24, 91–93 (1976)

    MathSciNet  MATH  Google Scholar 

  27. Skriganov, M.: Proof of the Bethe–Sommerfeld conjecture in dimension two. Sov. Math. Dokl. 20, 89–90 (1979)

    MATH  Google Scholar 

  28. Skriganov, M.: Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators. Proc. Steklov Math. Inst. 171, 3–122 (1984)

    Google Scholar 

  29. Skriganov, M.: The spectrum band structure of the three-dimensional Schrödinger operator with periodic potential. Inv. Math. 80, 107–121 (1985)

    Article  ADS  MATH  Google Scholar 

  30. Veliev, O.A.: Spectrum of multidimensional periodic operators. Teor. FunktsiĭFunktsional. Anal. i Prilozhen 49, 17–34 (1988). (in Russian)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

We are grateful to Leonid Parnovski for useful comments on an earlier version of this manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematics, Rice University, Houston, TX, 77005, USA

    David Damanik

  2. Department of Mathematics, Virginia Tech, 225 Stanger Street, Blacksburg, VA, 24061, USA

    Jake Fillman

  3. Department of Mathematics, University of California, Irvine, CA, 92697, USA

    Anton Gorodetski

Authors
  1. David Damanik
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jake Fillman
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Anton Gorodetski
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jake Fillman.

Additional information

Communicated by Alain Joye.

Publisher's Note

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

David Damanik was supported in part by NSF Grant DMS–1700131. Jake Fillman was supported in part by an AMS-Simons Travel Grant, 2016–2018. Anton Gorodetski was supported in part by Simons Fellowship (Grant Number 556910).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Damanik, D., Fillman, J. & Gorodetski, A. Multidimensional Almost-Periodic Schrödinger Operators with Cantor Spectrum. Ann. Henri Poincaré 20, 1393–1402 (2019). https://doi.org/10.1007/s00023-019-00768-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00023-019-00768-5

Mathematics Subject Classification

Search

Navigation