Communications in Mathematical Physics

, Volume 264, Issue 2, pp 371–389

Absolutely Continuous Spectra of Quantum Tree Graphs with Weak Disorder

  • Michael Aizenman
  • Robert Sims
  • Simone Warzel
Article

DOI: 10.1007/s00220-005-1468-5

Cite this article as:
Aizenman, M., Sims, R. & Warzel, S. Commun. Math. Phys. (2006) 264: 371. doi:10.1007/s00220-005-1468-5

Abstract

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.

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