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