Lyapounov exponent of the one dimensional Anderson model : weak disorder expansions

Abstract

We describe a method which gives the weak disorder expansion (λ → 0) of the Lyapounov exponent γ(E) of a discretized one-dimensional Schrödinger equation ψn+1 + ψ n-1 + λVnψn = Eψn with a random potential Vn. Near the band edge of the pure system (E → 2), the weak disorder expansion of y(E) is non analytic and we show that γ(E) ∼ λ2/3 when λ → 0. At the band centre (E → 0), we recover the anomaly which has already been explained by Kappus and Wegner. We find another anomaly at the energy E = 2 cos (π/3) and we believe that similar anomalies should occur at all energies E = 2 cos (απ) with α rational