Algebraic delocalization for the Schr\"odinger equation on large tori

TL;DR

This paper investigates how solutions to the Schr"odinger equation on large tori become delocalized as the system size grows, revealing algebraic bounds on localization and implications for disordered systems like the Anderson-Bernoulli model.

Contribution

It establishes an algebraic delocalization theorem for Schr"odinger solutions on large tori, connecting spectral properties with localization behavior in disordered systems.

Findings

Probability measures cannot be localized inside small balls unless decay is slow.

Provides algebraic bounds on localization length as energy approaches zero.

Implications for the Anderson-Bernoulli model showing algebraic blow-up of localization length.

Abstract

Let $\mathcal{L}$ be a fixed $d$-dimensional lattice. We study the localization properties of solutions of the stationary Schr\"odinger equation with a positive $L^\infty$ potential on tori $\mathbb{R}^d/L\mathcal{L}$ in the limit, as $L\to\infty$, for dimension $d \leq 3$. 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 $r = o(E^{-1/4+\epsilon})$, unless localization occurs with a sufficiently slow algebraic decay. In particular, we apply our result to Schr\"odinger 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 $d \geq 2$, and show that our theorem implies an algebraic blow-up of localization length in this limit.