Quantitative equidistribution of eigenfunctions for toral Schr\"odinger operators

We prove a quantum ergodicity theorem in position space for the eigenfunctions of a Schr\"odinger operator $-\Delta+V$ on a rectangular torus $\mathbb{T}^2$ for $V\in L^2(\mathbb{T}^2)$ 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 Schr\"odinger operator whose potential models disordered systems with $N$ obstacles. We prove the validity of this equidistribution theorem in the limit, as $N\to\infty$, 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 $N\to\infty$) this corresponds to a size decaying regime, as the coupling parameter in front of the potential will decay, as $N\to\infty$. We apply our result to Schr\"odinger operators modeling disordered systems on large tori $\mathbb{T}^2_L$ by scaling back to the fixed torus $\mathbb{T}^2$. In the case of random Schr\"odinger 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, nonzero 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.