Letters in Mathematical Physics

, Volume 106, Issue 10, pp 1317–1343 | Cite as

Spectra of Semi-Infinite Quantum Graph Tubes

  • Stephen P. Shipman
  • Jeremy Tillay


The spectrum of a semi-infinite quantum graph tube with square period cells is analyzed. The structure is obtained by rolling up a doubly periodic quantum graph into a tube along a period vector and then retaining only a semi-infinite half of the tube. The eigenfunctions associated to the spectrum of the half-tube involve all Floquet modes of the full tube. This requires solving the complex dispersion relation \({D(\lambda,k_1,k_2)=0}\) with \({(k_1,k_2)\in(\mathbb{C}/2\pi\mathbb{Z})^2}\) subject to the constraint \({a k_1 + b k_2 \equiv 0}\) (mod \({2\pi}\)), where a and b are integers. The number of Floquet modes for a given \({\lambda\in\mathbb{R}}\)  is  \({2\max\left\{ a, b \right\}}\). Rightward and leftward modes are determined according to an indefinite energy flux form. The spectrum may contain eigenvalues that depend on the boundary conditions, and some eigenvalues may be embedded in the continuous spectrum.


quantum graph spectrum Floquet modes embedded eigenvalue nanotube complex dispersion relation 

Mathematics Subject Classification

34A33 34B09 34B45 34B60 34L10 34L25 47A75 47B25 81U30 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aya, H., Cano, R., Zhevandrov, P.: Scattering and Embedded Trapped Modes for an Infinite Nonhomogeneous Timoshenko beam. Kluwer Academic Publishers, The Netherlands (2012)Google Scholar
  2. 2.
    Berkolaiko, G., Kuchment, P.: Introduction to Quantum Graphs. Mathematical Surveys and Monographs, vol. 186. AMS, Providence (2013)Google Scholar
  3. 3.
    Cattaneo C.: The spectrum of the continuous Laplacian on a graph. Monatshefte für Mathematik 124(3), 215–235 (1997)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Chihara, T.S.: An Introduction to Orthogonal Polynomials. Gordon and Breach, New York (1978)Google Scholar
  5. 5.
    Chung, F.: Spectral Graph Theory. American Mathematical Society, Providence (1997)Google Scholar
  6. 6.
    de Verdière, Y.C.: Spectres de graphes. Societe Mathematique de France. Cours specialises, vol 4. Société mathématique de France (1998)Google Scholar
  7. 7.
    Gohberg, I., Lancaster, P., Rodman, L.: Indefinite Linear Algebra and Applications. Birkhäuser Verlag AG, Switzerland (2005)Google Scholar
  8. 8.
    Goldstein C.I.: Eigenfunction expansions associated with the Laplacian for certain domains with infinite boundaries. I. Trans. Am. Math. Soc. 135, 1–31 (1969)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Iantchenko A., Korotyaev E.: Schrödinger operator on the zigzag half-nanotube in magnetic field. Math. Model. Nat. Phenom. 5(4), 175–197 (2010)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Korotyaev, E., Lobanov, I.: Zigzag periodic nanotube in magnetic field (2006). arXiv:math/0604007v1
  11. 11.
    Korotyaev E., Lobanov I.: Schrödinger operators on zigzag nanotubes. Annales Henri Poincaré 8(6), 1151–1176 (2007)ADSMathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Kuchment P.: Quantum graphs II. Some spectral properties of quantum and combinatorial graphs. J. Phys. A 38, 4887–4900 (2005)ADSMathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kuchment P., Post O.: On the spectra of carbon nano-structures. Commun. Math. Phys. 275(3), 805–826 (2007)ADSMathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Lambin P., Meunier V.: Structural properties of junctions between two carbon nanotubes. Appl. Phys. A 68(3), 263–266 (1999)ADSCrossRefGoogle Scholar
  15. 15.
    Louie, S.G.: Carbon nanotubes: synthesis, structure, properties, and applications. In: Electronic Properties, Junctions, and Defects of Carbon Nanotubes, pp. 113–145. Springer Berlin (2001)Google Scholar
  16. 16.
    Reed, M., Simon, B.: Methods of Mathematical Physics: Analysis of Operators, vol. IV. Academic Press, New York (1980)Google Scholar
  17. 17.
    Shipman S.P.: Eigenfunctions of unbounded support for embedded eigenvalues of locally perturbed periodic graph operators. Commun. Math. Phys. 332(2), 605–626 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Shipman S.P., Ribbeck J., Smith K.H., Weeks C.: A discrete model for resonance near embedded bound states. IEEE Photonics J. 2(6), 911–923 (2010)CrossRefGoogle Scholar
  19. 19.
    Shipman, S.P., Welters, A.T.: Resonance in anisotropic layered media. In: Proceedings of the International Conference on Mathematical Methods in EM Theory, pp. 227–232, Kharkov, IEEE (2012)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of MathematicsLouisiana State UniversityBaton RougeUSA
  2. 2.Department of Computational and Applied MathematicsRice UniversityHoustonUSA

Personalised recommendations