On the spectral theory of operators on trees

Abstract

We study a class of rooted trees with a substitution type structure. These trees are not necessarily regular, but exhibit a lot of symmetries. We consider nearest neighbor operators which reflect the symmetries of the trees. The spectrum of such operators is proven to be purely absolutely continuous and to consist of finitely many intervals. We further investigate stability of the absolutely continuous spectrum under perturbations by sufficiently small potentials. On the one hand, we look at a class of deterministic potentials which include radial symmetric ones. The absolutely continuous spectrum is stable under sufficiently small perturbations of this type if and only if the tree is not regular. On the other hand, we study random potentials. In this case, we prove stability of absolutely continuous spectrum for both regular and non regular trees provided the potentials are sufficiently small.