On absence of diffusion near the bottom of the spectrum for a random Schrödinger operator onL ²(ℝ)⁺

Abstract

We consider a random Schrödinger operator onL ²(ℝ^v) 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 Fröhlich and Spencer for the analysis of the Anderson model.