Skip to main content
SpringerLink
Log in

Discrete Bethe–Sommerfeld Conjecture

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

In this paper, we prove a discrete version of the Bethe–Sommerfeld conjecture. Namely, we show that the spectra of multi-dimensional discrete periodic Schrödinger operators on \({\mathbb{Z}^d}\) lattice with sufficiently small potentials contain at most two intervals. Moreover, the spectrum is a single interval, provided at least one of the periods is odd, and can have a gap whenever all periods are even.

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. Dahlberg B.E.J., Trubowitz E.: A remark on two dimensional periodic potentials. Comment. Math. Helvetici 57, 130–134 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  2. Embree, M., Fillman, J.: Spectra of discrete two-dimensional periodic Schrödinger operators with small potentials. J. Spectr. Theory. arXiv:1701.00863v1

  3. 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 

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

    Book  Google Scholar 

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

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Parnovski, L., Sobolev, A.V.: On the Bethe-Sommerfeld conjecture for the polyharmonic operator. Duke Math. J. 107, 209–238 (2001)

  8. Parnovski, L., Sobolev, A.V.: Lattice points, perturbation theory and the periodic polyharmonic operato. Ann. Henri Poincare 2, 573–581 (2001)

  9. 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) (Russian)

    MATH  Google Scholar 

  10. Skriganov M.: Proof of the Bethe–Sommerfeld conjecture in dimension two. Soviet Math. Dokl. 20(1), 89–90 (1979)

    MATH  Google Scholar 

  11. Skriganov, M.: Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators. Proc. Steklov Math. Inst. 171, 1–121 (1984)

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Veliev O.A.: Perturbation theory for the periodic multidimensional Schrödinger operator and the Bethe–Sommerfeld Conjecture. Int. J. Contemp. Math. Sci. 2(2), 19–87 (2007)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Advanced Study, Princeton, USA

    Rui Han

  2. University of California, Irvine, Irvine, USA

    Svetlana Jitomirskaya

Authors
  1. Rui Han
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Svetlana Jitomirskaya
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rui Han.

Additional information

Communicated by P. Deift

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Han, R., Jitomirskaya, S. Discrete Bethe–Sommerfeld Conjecture. Commun. Math. Phys. 361, 205–216 (2018). https://doi.org/10.1007/s00220-018-3141-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-018-3141-9

Search

Navigation