@article{b732b3ea32bb46d7a642746623bfb8db,

title = "Absolutely continuous spectrum for random operators on trees of finite cone type",

abstract = "We study the spectrum of random operators on a large class of trees. These trees have finitely many cone types and they can be constructed by a substitution rule. The random operators are perturbations of Laplace type operators either by random potentials or by random hopping terms, i. e., perturbations of the off-diagonal elements. We prove stability of arbitrary large parts of the absolutely continuous spectrum for sufficiently small but extensive disorder.",

author = "Matthias Keller and Daniel Lenz and Simone Warzel",

note = "Funding Information: This yields the absence of singular spectrum by a theorem of Klein [Kl1, Theorem 4.1]. Moreover, Im Gx(E + iη, Hλ,ω) can only tend to zero as η → 0 on subsets of I × of (Leb ⊗ P)-measure zero. Otherwise, this leads to a contradiction to Proposition 8. Hence we conclude I ⊆ σ(Hλ,ω) almost surely. □ Proof of Theorem 1. By Proposition 1, the set Σ consists of finitely many intervals, and clos Σ = σ(T ). Hence, there exists a finite set Σ0 such that Σ = σ(T ) \ Σ0. Therefore, the statement follows from Theorem 2. □ Acknowledgements. This work was started while M. Keller was visiting Princeton University and finished during his visit at the Hebrew University, where he was supported by the Israel Science Foundation (Grant no. 1105/10). He thanks the Department of Mathematics of both places for their hospitality.",

year = "2012",

month = oct,

doi = "10.1007/s11854-012-0040-4",

language = "English",

volume = "118",

pages = "363--396",

journal = "Journal d'Analyse Mathematique",

issn = "0021-7670",

publisher = "Springer New York",

number = "1",

}