Communications in Mathematical Physics

, Volume 264, Issue 2, pp 371–389 | Cite as

Absolutely Continuous Spectra of Quantum Tree Graphs with Weak Disorder

  • Michael Aizenman
  • Robert Sims
  • Simone Warzel


We consider the Laplacian on a rooted metric tree graph with branching number K≥2 and random edge lengths given by independent and identically distributed bounded variables. Our main result is the stability of the absolutely continuous spectrum for weak disorder. A useful tool in the discussion is a function which expresses a directional transmission amplitude to infinity and forms a generalization of the Weyl-Titchmarsh function to trees. The proof of the main result rests on upper bounds on the range of fluctuations of this quantity in the limit of weak disorder.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing 
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  1. 1.
    Acosta, V., Klein, A.: Analyticity of the density of states in the Anderson model in the Bethe lattice. J. Stat. Phys. 69, 277–305 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Aizenman, M., Sims, R., Warzel, S.: Stability of the absolutely continuous spectrum of random Schrödinger operators on tree graphs., 2005. To appear in Probab. Theor. Relat. FieldsGoogle Scholar
  3. 3.
    Carmona, R., Lacroix, J.: Spectral theory of random Schrödinger operators. Boston: Birkhäuser, 1990Google Scholar
  4. 4.
    Coddington, E.A., Levinson, N.: Theory of ordinary differential equations. New York: McGraw-Hill, 1955Google Scholar
  5. 5.
    Duren, P.L.: Theory of H p spaces. New York: Academic, 1970Google Scholar
  6. 6.
    Hupfer, T., Leschke, H., Müller, P., Warzel, S.: Existence and uniqueness of the integrated density of states for Schrödinger operators with magnetic fields and unbounded random potentials. Rev. Math. Phys. 13, 1547–1581 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Kostrykin, V., Schrader, R.: Kirchhoff's rule for quantum wires. J. Phys. A 32, 595–630 (1999)CrossRefADSzbMATHMathSciNetGoogle Scholar
  8. 8.
    Kostrykin, V., Schrader, R.: A random necklace model. Waves and random media 14, S75–S9032 (2004)Google Scholar
  9. 9.
    Kotani, S.: One-dimensional random Schrödinger operators and Herglotz functions. In: K. Ito (ed.), Taneguchi Symp. PMMP, Amsterdam: North Holland, 1985, pp. 219–250Google Scholar
  10. 10.
    Kottos, T., Smilansky, U.: Periodic orbit theory and spectral statistics for quantum graphs. Ann. Phys. 274, 76–124 (1999)CrossRefADSzbMATHMathSciNetGoogle Scholar
  11. 11.
    Kuchment, P.: Graph models for waves in thin structures. Waves and random media 12, R1–R24 (2002)Google Scholar
  12. 12.
    Kuchment, P.: Quantum graphs: I. Some basic structures. Waves and random media 14, S107–S128 (2004)Google Scholar
  13. 13.
    Kuchment, P.: Quantum graphs II. Some spectral properties of quantum and combinatorial graphs. preprint. J. Phys. A: Math. Gen. 38, 4887–4900 (2005)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Miller, J.D., Derrida, B.: Weak disorder expansion for the Anderson model on a tree. J. Stat. Phys. 75, 357–388 (1993)CrossRefGoogle Scholar
  15. 15.
    Minami, N.: An extension of Kotani's theorem to random generalized Sturm-Liouville operators. Commun. Math. Phys. 103, 387–402 (1986)CrossRefADSzbMATHMathSciNetGoogle Scholar
  16. 16.
    Minami, N.: An extension of Kotani's theorem to random generalized Sturm-Liouville operators II. In: Stochastic processes in classical and quantum systems, Lecture Notes in Physics 262, Berlin-Heidelberg-New York: Springer, 1986, pp. 411–419Google Scholar
  17. 17.
    Pastur, L., Figotin, A.: Spectra of random and almost-periodic operators. Berlin: Springer, 1992Google Scholar
  18. 18.
    Reed, M., Simon, B.: Methods of modern mathematical physics IV: Analysis of operators. New York: Academic Press, 1978Google Scholar
  19. 19.
    Schanz, H., Smilansky, U.: Periodic-orbit theory of Anderson-localization on graphs. Phys. Rev. Lett. 84, 1427–1430 (2000)CrossRefADSGoogle Scholar
  20. 20.
    Schenker, J.H., Aizenman, M.: The creation of spectral gaps by graph decorations. Lett. Math. Phys. 53, 253–262 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Sobolev, A.V., Solomyak, M.: Schrödinger operators on homogeneous metric trees: spectrum in gaps. Rev. Math. Phys. 14, 421–467 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Solomyak, M.: On the spectrum of the Laplacian on regular metric trees. Waves and random media 14, S155–S171 (2004)Google Scholar

Copyright information

© The authors 2005

Authors and Affiliations

  • Michael Aizenman
    • 1
  • Robert Sims
    • 2
  • Simone Warzel
    • 1
  1. 1.Departments of Mathematics and PhysicsPrinceton UniversityPrincetonUSA
  2. 2.Department of MathematicsUniversity of California at DavisDavisUSA

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