Algebraic delocalization for the Schrödinger equation on large tori

Let $\mathcal{L}$ be a fixed $d$-dimensional lattice. We study the localization properties of solutions of the stationary Schrödinger equation with a positive $L^{\mathrm{\infty }}$ potential on tori ${\mathbb{R}}^d\mathrm{/}L\mathcal{L}$ in the limit, as $L\to \mathrm{\infty }$, for dimension $d\le 3$. We show that the probability measures associated with $L^2$-normalized solutions, with eigenvalue $E$ near the bottom of the spectrum, satisfy an algebraic delocalization theorem which states that these probability measures cannot be localized inside a ball of radius $r=o(E^{-1\mathrm{/}4+\epsilon })$, unless localization occurs with a sufficiently slow algebraic decay. In particular, we apply our result to Schrödinger operators modeling disordered systems, such as the d-dimensional continuous Anderson- Bernoulli model, where almost sure exponential localization of eigenfunctions, in the limit as $E\to 0$, was proved by Bourgain-Kenig in dimension $d\ge 2$, and show that our theorem implies an algebraic blow-up of localization length in this limit.