Absolutely continuous spectra of quantum tree graphs with weak disorder

Michael Aizenman, Robert Sims, Simone Warzel

Research output: Contribution to journalArticlepeer-review

50 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)371-389
Number of pages19
JournalCommunications In Mathematical Physics
Issue number2
StatePublished - Jun 2006

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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