Transfer matrices, hyperbolic geometry and absolutely continuous spectrum for some discrete Schrödinger operators on graphs

We prove the existence of absolutely continuous spectrum for a class of discrete Schrödinger operators on tree like graphs. We consider potentials whose radial behaviour is subject only to an ∞ bound. In the transverse direction the potential must satisfy a condition such as periodicity. The graphs we consider include binary trees and graphs obtained from a binary tree by adding edges, possibly with weights. Our methods are motivated by the one-dimensional transfer matrix method, interpreted as a discrete dynamical system on the hyperbolic plane. This is extended to more general graphs, leading to a formula for the Green’s function. Bounds on the Green’s function then follow from the contraction properties of the transformations that arise in this generalization. The bounds imply the existence of absolutely continuous spectrum. © 2005 Elsevier Inc. All rights reserved.