Journal of Statistical Physics

, Volume 131, Issue 4, pp 651–673 | Cite as

Localization on Quantum Graphs with Random Vertex Couplings



We consider Schrödinger operators on a class of periodic quantum graphs with randomly distributed Kirchhoff coupling constants at all vertices. We obtain necessary conditions for localization on quantum graphs in terms of finite volume criteria for some energy-dependent discrete Hamiltonians. These conditions hold in the strong disorder limit and at the spectral edges.


Random operators Quantum graph Localization 


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  1. 1.
    Aizenman, M.: Localization at weak disorder: some elementary bounds. Rev. Math. Phys. 6, 1163–1182 (1994). Special issue CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Aizenman, M., Schenker, J.H., Friedrich, R.M., Hundertmark, D.: Finite-volume fractional-moment criteria for Anderson localization. Commun. Math. Phys. 224, 219–253 (2001) CrossRefADSMathSciNetMATHGoogle Scholar
  3. 3.
    Aizenman, M., Elgart, A., Naboko, S., Schenker, J.H., Stolz, G.: Moment analysis of localization in random Schroedinger operators. Invent. Math. 163, 343–413 (2006) CrossRefADSMathSciNetMATHGoogle Scholar
  4. 4.
    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
  5. 5.
    Amrein, W.O., Georgescu, V.: On the characterization of bound states and scattering states in quantum mechanics. Helvetica Phys. Acta 46, 635–658 (1973) MathSciNetGoogle Scholar
  6. 6.
    Boutet de Monvel, A., Grinshpun, V.: Exponential localization for multidimensional Schrödinger operators with random point potentials. Rev. Math. Phys. 9, 425–451 (1997) CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Brüning, J., Geyler, V., Pankrashkin, K.: Cantor and band spectra for periodic quantum graphs with magnetic fields. Commun. Math. Phys. 269, 87–105 (2007) CrossRefADSMATHGoogle 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). Preprint arXiv:math-ph/0611088 CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Dorlas, T.C., Macris, N., Pulé, J.V.: Characterization of the spectrum of the Landau Hamiltonian with delta impurities. Commun. Math. Phys. 204, 367–396 (1999) CrossRefADSMATHGoogle Scholar
  10. 10.
    Exner, P.: Lattice Kronig-Penney models. Phys. Rev. Lett. 74, 3503–3506 (1995) CrossRefADSGoogle Scholar
  11. 11.
    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) CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Geyler, V.A., Margulis, V.A.: Anderson localization in the nondiscrete Maryland model. Theor. Math. Phys. 70, 133–140 (1987) CrossRefGoogle Scholar
  13. 13.
    Gnutzmann, S., Smilansky, U.: Quantum graphs: applications to quantum chaos and universal spectral statistics. Adv. Phys. 55, 527–625 (2006) CrossRefADSGoogle Scholar
  14. 14.
    Gruber, M.J., Lenz, D., Veselić, I.: Uniform existence of the integrated density of states for random Schrödinger operators on metric graphs over ℤd. J. Funct. Anal. 253, 515–533 (2007). Preprint arXiv:math.SP/0612743 CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Helm, M., Veselić, I.: A linear Wegner estimate for alloy type Schrödinger operators on metric graphs. J. Math. Phys. 48, 092107 (2007) CrossRefADSMathSciNetGoogle Scholar
  16. 16.
    Hislop, P.D., Post, O.: Anderson localization for radial tree-like random quantum graphs. Preprint arXiv:math-ph/0611022 Google Scholar
  17. 17.
    Hislop, P.D., Kirsch, W., Krishna, M.: Spectral and dynamical properties of random models with non-local and singular interactions. Math. Nachr. 278, 627–664 (2005) CrossRefMathSciNetMATHGoogle Scholar
  18. 18.
    Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1966) MATHGoogle Scholar
  19. 19.
    Klopp, F.: Localization for semi-classical continuous random Schrödinger operators II: the random displacement model. Helvetica Phys. Acta 66, 810–841 (1993) MathSciNetMATHGoogle Scholar
  20. 20.
    Klopp, F.: Localisation pour des opérateurs de Schrödinger aléatoires dans L 2(R d): un modèle semi-classique. Ann. Inst. Fourier 45, 265–316 (1995) MathSciNetMATHGoogle Scholar
  21. 21.
    Klopp, F.: Band edge behaviour for the integrated density of states of random Jacobi matrices in dimension 1. J. Stat. Phys. 90, 927–947 (1998) CrossRefMathSciNetMATHADSGoogle Scholar
  22. 22.
    Klopp, F.: Weak disorder localization and Lifshitz tails. Commun. Math. Phys. 232, 125–155 (2002) CrossRefADSMathSciNetMATHGoogle Scholar
  23. 23.
    Klopp, F., Wolff, T.: Lifschitz tails for 2-dimensional random Schrödinger operators. J. Anal. Math. 88, 63–147 (2002) MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Kostrykin, V., Schrader, R.: A random necklace model. Waves Random Media 14, S75–S90 (2004) CrossRefADSMathSciNetMATHGoogle Scholar
  25. 25.
    Kuchment, P.: Quantum graphs I. Some basic structures. Waves Random Media 14, S107–S128 (2004) CrossRefADSMathSciNetMATHGoogle Scholar
  26. 26.
    Kuchment, P.: Quantum graphs II. Some spectral properties of quantum and combinatorial graphs. J. Phys. A: Math. Gen. 38, 4887–4900 (2005) CrossRefADSMathSciNetMATHGoogle Scholar
  27. 27.
    Kunz, H., Souillard, B.: Sur le spectre des opérateurs aux différences finies aléatoires. Commun. Math. Phys. 78, 201–246 (1980) CrossRefADSMathSciNetMATHGoogle Scholar
  28. 28.
    Levitan, B.M., Sargsyan, I.S.: Sturm-Liouville and Dirac Operators. Kluwer, Dordrecht (1990) MATHGoogle Scholar
  29. 29.
    Pankrashkin, K.: Localization effects in a periodic quantum graph with magnetic field and spin-orbit interaction. J. Math. Phys. 47, 112105 (2006) CrossRefADSMathSciNetGoogle Scholar
  30. 30.
    Pankrashkin, K.: Spectra of Schrödinger operators on equilateral quantum graphs. Lett. Math. Phys. 77, 139–154 (2006) CrossRefMathSciNetMATHADSGoogle Scholar
  31. 31.
    Pastur, L., Figotin, A.: Spectra of Random and Almost-Periodic Operators. Springer, Berlin (1992) MATHGoogle Scholar
  32. 32.
    Posilicano, A.: Self-adjoint extensions of restrictions. Preprint arXiv:math-ph/0703078 Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.L.A.G.A., CNRS UMR 7539, Institut GaliléeUniversité Paris NordVilletaneuseFrance
  2. 2.Institut Universitaire de FranceParisFrance
  3. 3.Institut für MathematikHumboldt-Universität zu BerlinBerlinGermany

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