Localization Lengths for Schrödinger Operators on Z with Decaying Random Potentials

Abstract. We study a class of Schrödinger operators on Z with a random potential decaying as |x|, 0 < σ ≤ 1 2 , in the limit of small disorder strength λ. For the critical exponent σ = 1 2 , we prove that the localization length of eigenfunctions is bounded below by 2 − 1 4 +η , while for 0 < σ < 1 2 , the lower bound is λ− 2−η 1−2σ , for any η > 0. These estimates ”interpolate” between the lower bound λ due to recent work of Schlag-Shubin-Wolff for σ = 0, and pure a.c. spectrum for σ > 1 2 demonstrated in recent work of Bourgain.