Absolutely Continuous Spectra of Quantum Tree Graphs with Weak Disorder

Details Absolutely Continuous Spectra of Quantum Tree Graphs with Weak Disorder Absolutely Continuous Spectra of Quantum Tree Graphs with Weak Disorder

2005-04-12

arxiv.org/abs/math-ph/0504039v3

Commun. Math. Phys. 264 (2006) 371 - 389

Physics

Mathematical Physics

Scientific paper

10.1007/s00220-005-1468-5

We consider the Laplacian on a rooted metric tree graph with branching number $ K \geq 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.

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