Absolutely Continuous Spectra of Quantum Tree Graphs with Weak Disorder
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- Aizenman, M., Sims, R. & Warzel, S. Commun. Math. Phys. (2006) 264: 371. doi:10.1007/s00220-005-1468-5
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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.