WEAK-DISORDER EXPANSION FOR THE ANDERSON MODEL ON A TREE

We show how certain properties of the Anderson model on a tree are rel ated to the solutions of a nonlinear integral equation. Whether the wa ve function is extended or localized, for example, corresponds to whet her or not the equation has a complex solution. We show how the equati on can be solved in a weak-disorder expansion. We find that, for small disorder strength lambda, there is an energy E(c)(lambda) above which the density of states and the conducting properties vanish to alt ord ers in perturbation theory. We compute perturbatively the position of the line E(c)(lambda) which begins, in the limit of zero disorder, at the band edge of the pure system. Inside the band of the pure system t he density of states and conducting properties can be computed perturb atively. This expansion breaks down near E(c)(lambda) because of small denominators. We show how it can be resummed by choosing the appropri ate scaling of the energy. For energies greater than E(c)(lambda) we s how that nonperturbative effects contribute to the density of states b ut we have been unable to tell whether they also contribute to the con ducting properties.