Skip to main content
SpringerLink
Log in

Invariants for Laplacians on periodic graphs

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

We consider a Laplacian on periodic discrete graphs. Its spectrum consists of a finite number of bands. In a class of periodic 1-forms, i.e., functions defined on edges of the periodic graph, we introduce a subclass of minimal forms with a minimal number \({{\mathcal {I}}}\) of edges in their supports on the period. We obtain a specific decomposition of the Laplacian into a direct integral in terms of minimal forms, where fiber Laplacians (matrices) have the minimal number \(2{{\mathcal {I}}}\) of coefficients depending on the quasimomentum and show that the number \({{\mathcal {I}}}\) is an invariant of the periodic graph. Using this decomposition, we estimate the position of each band, the Lebesgue measure of the Laplacian spectrum and the effective masses at the bottom of the spectrum in terms of the invariant \({{\mathcal {I}}}\) and the minimal forms. In addition, we consider an inverse problem: we determine necessary and sufficient conditions for matrices depending on the quasimomentum on a finite graph to be fiber Laplacians. Moreover, similar results for Schrödinger operators with periodic potentials are obtained.

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.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Biggs, N.: Algebraic Graph Theory. Cambridge University Press, Cambridge (1974)

    Book  MATH  Google Scholar 

  2. Cvetkovic, D., Doob, M., Sachs, H.: Spectra of Graphs. Theory and Applications. Johann Ambrosius Barth, Heidelberg (1995)

    MATH  Google Scholar 

  3. Fabila-Carrasco, J.S., Lledó, F., Post, O.: Spectral gaps and discrete magnetic Laplacians. Linear Algebra Appl. 547(15), 183–216 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  4. Harris, P.: Carbon Nano-tubes and Related Structure. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  5. Higuchi, Y., Nomura, Y.: Spectral structure of the Laplacian on a covering graph. Eur. J. Combin. 30(2), 570–585 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Higuchi, Y., Shirai, T.: The spectrum of magnetic Schrödinger operators on a graph with periodic structure. J. Funct. Anal. 169, 456–480 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Higuchi, Y., Shirai, T.: A remark on the spectrum of magnetic Laplacian on a graph, the proceedings of TGT10. Yokohama Math. J. 47, 129–142 (1999). Special issue

    MathSciNet  MATH  Google Scholar 

  8. Higuchi, Y., Shirai, T.: Some spectral and geometric properties for infinite graphs. AMS Contemp. Math. 347, 29–56 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Korotyaev, E.: Estimates for the Hill operator. I. J. Differ. Equ. 162(1), 1–26 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Korotyaev, E.: Estimates for the Hill operator. II. J. Differ. Equ. 223(2), 229–260 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Korotyaev, E.: Effective masses for zigzag nanotubes in magnetic fields. Lett. Math. Phys. 83(1), 83–95 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Korotyaev, E., Saburova, N.: Schrödinger operators on periodic discrete graphs. J. Math. Anal. Appl. 420(1), 576–611 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Korotyaev, E., Saburova, N.: Spectral band localization for Schrödinger operators on periodic graphs. Proc. Am. Math. Soc. 143, 3951–3967 (2015)

    Article  MATH  Google Scholar 

  14. Korotyaev, E., Saburova, N.: Effective masses for Laplacians on periodic graphs. J. Math. Anal. Appl. 436(1), 104–130 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  15. Korotyaev, E., Saburova, N.: Magnetic Schrödinger operators on periodic discrete graphs. J. Funct. Anal. 272, 1625–1660 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. Korotyaev, E., Slousch, V.: Asymptotics and estimates of the discrete spectrum of the Schrödinger operator on a discrete periodic graph. preprint arXiv:1903.11810

  17. Kotani, M., Shirai, T., Sunada, T.: Asymptotic behavior of the transition probability of a random walk on an infinite graph. J. Funct. Anal. 159(2), 664–689 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lledó, F., Post, O.: Eigenvalue bracketing for discrete and metric graphs. J. Math. Anal. Appl. 348, 806–833 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Mohar, B.: Some relations between analytic and geometric properties of infinite graphs. Discrete Math. 95, 193–219 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  20. Novoselov, K.S., Geim, A.K., et al.: Electric field effect in atomically thin carbon films. Science 306(5696), 666–669 (2004)

    Article  Google Scholar 

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

    MATH  Google Scholar 

  22. Sunada, T.: Topological Crystallography, Surveys Tutorials Appl. Math. Sci., vol. 6. Springer, Tokyo (2013)

    Google Scholar 

  23. Sy, P.W., Sunada, T.: Discrete Schrödinger operator on a graph. Nagoya Math. J. 125, 141–150 (1992)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Our study was supported by the RSF grant No. 18-11-00032. We would like to thank a referee for thoughtful comments that helped us to improve the manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematical Analysis, Saint-Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg, 199034, Russia

    Evgeny Korotyaev

  2. Department of Mathematical Analysis, Algebra and Geometry, Northern (Arctic) Federal University, Severnaya Dvina emb. 17, Arkhangelsk, 163002, Russia

    Natalia Saburova

Authors
  1. Evgeny Korotyaev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Natalia Saburova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Natalia Saburova.

Additional information

Communicated by Y. Giga.

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Korotyaev, E., Saburova, N. Invariants for Laplacians on periodic graphs. Math. Ann. 377, 723–758 (2020). https://doi.org/10.1007/s00208-019-01842-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-019-01842-3

Search

Navigation