Letters in Mathematical Physics

, Volume 87, Issue 1–2, pp 99–114 | Cite as

Localization on Quantum Graphs with Random Edge Lengths

  • Frédéric Klopp
  • Konstantin Pankrashkin


The spectral properties of the Laplacian on a class of quantum graphs with random metric structure are studied. Namely, we consider quantum graphs spanned by the simple \({\mathbb Z^d}\) -lattice with δ-type boundary conditions at the vertices, and we assume that the edge lengths are randomly independently identically distributed. Under the assumption that the coupling constant at the vertices does not vanish, we show that the operator exhibits the Anderson localization near the spectral edges situated outside a certain forbidden set.

Mathematics Subject Classification (2000)

81Q10 35R60 47B80 60H25 


Quantum graph Random operator Random metric Anderson localization 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aizenman M., Sims R., Warzel S.: Absolutely continuous spectra of quantum tree graphs with weak disorder. Commun. Math. Phys. 264, 371–389 (2006)CrossRefADSMathSciNetGoogle Scholar
  2. 2.
    Albeverio S., Gesztesy F., Høegh-Krohn R., Holden H.: Solvable Models in Quantum Mechanics (with an appendix by P. Exner), 2nd edn. AMS, Providence (2005)Google Scholar
  3. 3.
    Aizenman M., Molchanov S.: Localization at large disorder and at extreme energies: an elementary derivation. Commun. Math. Phys. 157, 245–278 (1993)MATHCrossRefADSMathSciNetGoogle Scholar
  4. 4.
    Anderson P.: Absence of diffusion in certain random latices. Phys. Rev. 109, 1492–1505 (1958)CrossRefADSGoogle Scholar
  5. 5.
    von Below J.: A characteristic equation associated to an eigenvalue problem on c 2-networks. Linear Algebra Appl. 71, 309–325 (1985)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Berkolaiko, G., Carlson, R., Fulling, S.A., Kuchment, P.: (eds.) Quantum graphs and their applications. Contemp. Math. (AMS) 415 (2006)Google Scholar
  7. 7.
    Boutet de Monvel, A., Lenz, D., Stollmann, P.: Sch’nol’s theorem for strongly local forms. Israel J. Math. (to appear) (preprint arXiv:0708.1501)Google Scholar
  8. 8.
    Brüning J., Geyler V., Pankrashkin K.: Spectra of self-adjoint extensions and applications to solvable Schrödinger operators. Rev. Math. Phys. 20, 1–70 (2008)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Carmona R., Lacroix J.: Spectral Theory of Random Schrödinger Operators. Birkhäuser, Boston (1990)MATHGoogle Scholar
  10. 10.
    Chen, K., Molchanov, S., Vainberg, B.: Localization on Avron-Exner-Last graphs: I. Local perturbations. In [6] pp. 81–92Google Scholar
  11. 11.
    Disertori, M., Kirsch, W., Klein, A., Klopp, F., Rivasseau, V.: Random Schrödinger operators. Panoramas Synthèses, vol. 25. Soc. Math. France (2008)Google Scholar
  12. 12.
    Exner P.: A duality between Schrödinger operators on graphs and certain Jacobi matrices. Ann. Inst. Henri Poincaré Phys. Théor. 66, 359–371 (1997)MATHMathSciNetGoogle Scholar
  13. 13.
    Exner, P., Dell’Antonio, G., Geyler, V.: (eds.) Special Issue on Singular interactions in quantum mechanics: solvable models. J. Phys. A 38(22) (2005)Google Scholar
  14. 14.
    Exner P., Helm M., Stollmann P.: Localization on a quantum graph with a random potential on the edges. Rev. Math. Phys. 19, 923–939 (2007)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Exner, P., Keating, J.P., Kuchment, P., Sunada, T., Teplyaev, A.: (eds.) Analysis on graphs and its applications. In: Proc. Symp. Pure Math., vol. 77. AMS, Providence (2008)Google Scholar
  16. 16.
    Fröhlich J., Spencer T.: Absence of diffusion in the Anderson tight binding model. Commun. Math. Phys. 88, 151–184 (1983)MATHCrossRefADSGoogle Scholar
  17. 17.
    Ghribi, F., Hislop, P.D., Klopp, F.: Localization for Schrödinger operators with random vector potentials. In: Germinet, F., Hislop, P.D.: (eds.) Adventures in Mathematical Physics. Contemp. Math., vol. 447, pp. 123–138. AMS, Providence (2007)Google Scholar
  18. 18.
    Gnutzmann S., Smilansky U.: Quantum graphs: applications to quantum chaos and universal spectral statistics. Adv. Phys. 55, 527–625 (2006)CrossRefADSGoogle Scholar
  19. 19.
    Grenkova L., Molčanov S., Sudarev Y.: On the basic states of one-dimensional disordered structures. Commun. Math. Phys. 90, 101–123 (1983)MATHCrossRefADSGoogle Scholar
  20. 20.
    Gruber, M., Lenz, D., Veselić’, I.: Uniform existence of the integrated density of states for combinatorial and metric graphs over \({\mathbb Z^d}\) . In [15] pp. 97–108Google Scholar
  21. 21.
    Gruber, M., Helm, M., Veselić, I.: Optimal Wegner estimates for random Schrödinger operators on metric graphs. In [15] pp. 409–422Google Scholar
  22. 22.
    Hislop, P.D., Post, O.: Anderson localization for radial tree-like random quantum graphs (Preprint arXiv:math-ph/0611022), to appear in Waves Complex Random MediaGoogle Scholar
  23. 23.
    Klein, A.: Multiscale analysis and localization of random operators. In [11] pp. 1–39Google Scholar
  24. 24.
    Klopp F., Nakamura S.: A note on Anderson localization for the random hopping model. J. Math. Phys. 44, 4975–4980 (2003)MATHCrossRefADSMathSciNetGoogle Scholar
  25. 25.
    Klopp F., Pankrashkin K.: Localization on quantum graphs with random vertex couplings. J. Stat. Phys. 131, 651–673 (2008)MATHCrossRefADSMathSciNetGoogle Scholar
  26. 26.
    Kostrykin, V., Schrader, R.: A random necklace model. In [27] pp. S75–S90Google Scholar
  27. 27.
    Kuchment, P.: (ed.) Quantum graphs special section. Waves Random Media 14(1) (2004)Google Scholar
  28. 28.
    Kuchment, P.: Quantum graphs I. Some basic structures. In [27] pp. S107–S128Google Scholar
  29. 29.
    Kuchment, P.: Quantum graphs II. Some spectral properties of quantum and combinatorial graphs. In [13] pp. 4887–4900Google Scholar
  30. 30.
    Lenz, D., Peyerimhoff, N., Post, O., Veselic’, I.: Continuity of the integrated density of states on random length metric graphs. (preprint arXiv:0811.4513)Google Scholar
  31. 31.
    Lenz D., Schubert C., Stollmann P.: Eigenfunction expansion for Schrödinger operators on metric graphs. Int. Equ. Oper. Theory 62, 541–553 (2008)CrossRefGoogle Scholar
  32. 32.
    Najar H.: Non-Lifshitz tails at the spectral bottom of some random operators. J. Stat. Phys. 130, 713–725 (2008)MATHCrossRefADSMathSciNetGoogle Scholar
  33. 33.
    Nakamura S.: Lifschitz tail for 2D discrete Schrödinger with random magnetic field. Ann. Henri Poincaré 1, 823–835 (2000)MATHCrossRefGoogle Scholar
  34. 34.
    Pankrashkin K.: Localization effects in a periodic quantum graph with magnetic field and spin-orbit interaction. J. Math. Phys. 47, 112105 (2006)CrossRefADSMathSciNetGoogle Scholar
  35. 35.
    Pankrashkin K.: Spectra of Schrödinger operators on equilateral quantum graphs. Lett. Math. Phys. 77, 139–154 (2006)MATHCrossRefADSMathSciNetGoogle Scholar
  36. 36.
    Post, O.: Equilateral quantum graphs and boundary triples. In [15] pp. 469–490Google Scholar
  37. 37.
    Pastur L., Figotin A.: Spectra of Random and Almost-Periodic Operators. Springer, Berlin (1992)MATHGoogle Scholar
  38. 38.
    Stollmann P.: Caught by Disorder. Bound States in Random Media. Birkhäuser, Boston (2001)MATHGoogle Scholar
  39. 39.
    Vidal J., Butaud R., Douçot B., Mosseri R.: Disorder and interactions in Aharonov–Bohm cages. Phys. Rev. B 64, 155306 (2001)CrossRefADSGoogle Scholar

Copyright information

© Springer 2009

Authors and Affiliations

  1. 1.Laboratoire Analyse, Géométrie et Applications, CNRS UMR 7539, Institut GaliléeUniversité Paris-NordVilletaneuseFrance
  2. 2.Institut Universitaire de FranceParisFrance
  3. 3.Institut für MathematikHumboldt-UniversitätBerlinGermany
  4. 4.Laboratoire de mathématiques, CNRS UMR 8628Unversité Paris-SudOrsayFrance

Personalised recommendations