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

B. Derrida , E. Gardner

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.

Mots clés: Anderson model; band theory models and calculation methods; Schrodinger equation

Domaines: Articles anciens

B. Derrida, E. Gardner. Lyapounov exponent of the one dimensional Anderson model : weak disorder expansions. Journal de Physique, 1984, 45 (8), pp.1283-1295. ⟨10.1051/jphys:019840045080128300⟩. ⟨jpa-00209867⟩