Localization Lengths and Boltzmann Limit for the Anderson Model at Small Disorders in Dimension 3

We prove lower bounds on the localization length of eigenfunctions in the three-dimensional Anderson model at weak disorders. Our results are similar to those obtained by Schlag, Shubin and Wolff, J. Anal. Math. 88 (2002)], for dimensions one and two. We prove that with probability one, most eigenfunctions have localization lengths bounded from below by , where λ is the disorder strength. This is achieved by time-dependent methods which generalize those developed by Erdös and Yau Commun. Pure Appl. Math. LIII: 667–753 (2003)] to the lattice and non-Gaussian case. In addition, we show that the macroscopic limit of the corresponding lattice random Schrödinger dynamics is governed by a linear Boltzmann equation.