Contemporary Mathematics
Volume 412, 2006
Fluctuation based proof of the stability of ac spectra of
random operators on tree graphs
Michael Aizenman, Robert Sims, and Simone Warzel
ABSTRACT. We summarize recent works on the stability under disorder of the
absolutely continuous spectra of random operators on tree graphs. The cases
covered include: SchrOdinger operators with random potential, quantum graph
operators for trees with randomized edge lengths, and radial quasi-periodic
operators perturbed by random potentials.
1. Introduction
The incorporation of disorder is known to have strong effects on the spectral
properties of self adjoint operators and the dynamics they generate. The phenom-
enon is relevant for models of physics, where it carries significant implications for
the conduction properties of metals, the quantum hall effect, and the properties of
quantum networks. As has been shown in various disciplines, disorder is a fruitful
subject for mathematics. Yet, our understanding of its implications for random
operators is still underdeveloped. This is exemplified by the fact that there is, at
present, no example of a local operator in a finite dimensional space for which the
existence of continuous spectrum has been established in the presence of weak but
extensive, i.e., homogeneously random, disorder. Until recently, the one case for
which there has been a constructive result, due to A. Klein [K95, K98], is the dis-
crete Schrodinger operator with an iid random potential on a regular Bethe lattice.
Our recent work presents another method, with a different range of applicability,
for the proof of the stability, under weak disorder, of ac spectra of tree operators.
In this article we summarize the results and the main ideas which play a role in
their derivation.
1.1. Examples of operators with disorder.
A guiding example of a local
generator of quantum dynamics is the Schrodinger operator
Ho
=
-~+U
(1.1)
1991 Mathematics Subject Classification. Primary 82B44; Secondary 47B80.
Key words and phrases. Random SchrOdinger operators, ac spectrum, tree graphs.
The article covers talks given at UAB International Conference on Differential Equations and
Mathematical Physics, Birmingham, March 2005 (RS and SW}, and at AMS Snowbird Conference,
Utah, June 2005 (MA).
@2006 by the authors. Faithful reproduction for non-commercial purposes is permitted.
1
http://dx.doi.org/10.1090/conm/412/07763
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