Michael Aizenman, Robert Sims, Simone Warzel
Absolutely Continuous Spectra of Quantum Tree Graphs with Weak Disorder
(231K, pdf)
ABSTRACT. 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.