Quantitative equidistribution of eigenfunctions for toral Schrödinger operators

Abstract

We prove a quantum ergodicity theorem in position space for the eigenfunctions of a Schrödinger operator [Formula: see text] on a rectangular torus [Formula: see text] for [Formula: see text] with an algebraic rate of convergence in terms of the eigenvalue. A key application of our theorem is a quantitative equidistribution theorem for the eigenfunctions of a Schrödinger operator whose potential models disordered systems with [Formula: see text] obstacles. We prove the validity of this equidistribution theorem in the limit, as [Formula: see text], under the assumption that a weak overlap hypothesis is satisfied by the potentials modeling the obstacles, and we note that, when rescaling to a large torus (such that the density remains finite, as [Formula: see text]) this corresponds to a size decaying regime, as the coupling parameter in front of the potential will decay, as [Formula: see text]. We apply our result to Schrödinger operators modeling disordered systems on large tori [Formula: see text] by scaling back to the fixed torus [Formula: see text]. In the case of random Schrödinger operators, such as random displacement models, we deduce an almost sure equidistribution theorem on certain length scales which depend on the coupling parameter, the density of the potentials and the eigenvalue. In particular, if these parameters converge to finite, non-zero values, we are able to determine at which length scale (as a function of these parameters) equidistribution breaks down. In this sense, we provide a lower bound for the Anderson localization length as a function of energy, coupling parameter and the density of scatterers.