On absence of diffusion near the bottom of the spectrum for a random Schr�dinger operator onL 2(?)+

Abstract

We consider a random Schrodinger operator onL 2(ℝv) of the form $$H_\omega = - \Delta + V_\omega ,V_\omega (x) = \Sigma \chi _{C_i } (x)q_i (\omega )$$ , {C i} being a covering of ℝ v with unit cubes around the sites of ℤ v and {q i} i.i.d. random variables with values in [0, 1]. We assume that theq i's are continuously distributed with bounded densityf(q) and that 0<P(q 0<1/2)=α<1. Then we show that an ergodic mean of the quantity 〈∫dx|x|2|(exp(itH ω)Φ)(x)|2〉t −1 vanishes provided Φ=g E(H ω)Ψ, where Ψ is well-localized around the origin andg E is a positiveC ∞-function with support in (0,E),E≦E*(α, |f|∞). Our estimate ofE*(α, |f|∞) is such that the set {x∈ℝ v |V ∞(x) ≦E*(α, |f|∞)} may contain with probability one an infinite cluster of cubes {C i} which are nearest neighbours. The proof is based on the technique introduced by Frohlich and Spencer for the analysis of the Anderson model.