Integral Equations and Operator Theory

, Volume 81, Issue 4, pp 535–557 | Cite as

Spectrum of a Dilated Honeycomb Network

Article

Abstract

We analyze spectrum of Laplacian supported by a periodic honeycomb lattice with generally unequal edge lengths and a δ type coupling in the vertices. Such a quantum graph has nonempty point spectrum with compactly supported eigenfunctions provided all the edge lengths are commensurate. We derive conditions determining the continuous spectral component and show that existence of gaps may depend on number-theoretic properties of edge lengths ratios. The case when two of the three lengths coincide is discussed in detail.

Mathematics Subject Classification

Primary 81Q35 Secondary 34B45 34K13 35B10 

Keywords

Quantum graphs Hexagon lattice Laplace operator Vertex δ-coupling Spectrum 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Band R., Berkolaiko G.: Universality of the momentum band density of periodic networks. Phys. Rev. Lett. 111, 130404 (2013)CrossRefGoogle Scholar
  2. 2.
    Berkolaiko, G., Kuchment, P.: Introduction to Quantum Graphs. Amer. Math. Soc., Providence (2013)Google Scholar
  3. 3.
    Exner P.: Contact interactions on graph superlattices. J. Phys. A: Math. Gen. 29, 87–102 (1996)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Exner P., Gawlista R.: Band spectra of rectangular graph superlattices. Phys. Rev. B 53, 7275–7286 (1996)CrossRefGoogle Scholar
  5. 5.
    Kuchment P., private communication following the publication of [3]Google Scholar
  6. 6.
    Kuchment P.: Quantum graphs: II. Some spectral properties of quantum and combinatorial graphs. J. Phys. A.: Math. Gen. 38, 4887–4900 (2005)MATHMathSciNetGoogle Scholar
  7. 7.
    Kuchment P., Post O.: On the spectra of carbon nano-structures. Commun. Math. Phys. 275(3), 805–826 (2007)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Last Y.: Zero measure spectrum for almost Mathieu operator. Commun. Math. Phys. 164, 421–432 (1994)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Schanz H., Kottos T.: Scars on quantum networks ignore the Lyapunov exponent. Phys. Rev. Lett. 90, 234101 (2003)CrossRefGoogle Scholar
  10. 10.
    Schmidt W.M.: Diophantine Approximations and Diophantine Equations Lecture Notes in Mathematics, vol. 1467. Springer, Berlin (1991)Google Scholar
  11. 11.
    de Verdière Y.C.: Semi-classical measures on Quantum Graphs and the Gauss map of the determinant manifold (preprint). arXiv:1311.5449
  12. 12.
    Veselic I.: Spectral analysis of percolation Hamiltonians. Math. Ann. 331, 841–865 (2005)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Nuclear Physics Institute, Academy of Sciences of the Czech RepublicŘežCzech Republic
  2. 2.Doppler InstituteCzech Technical UniversityPragueCzech Republic
  3. 3.Bogolyubov Laboratory of Theoretical Physics, Joint Institute for Nuclear ResearchDubnaRussia

Personalised recommendations