Uniformly distributed eigenfunctions on tori with random impurities

TL;DR

This paper investigates conditions under which eigenfunctions of a random Schrödinger operator on large tori with random impurities are uniformly distributed, revealing a link between system parameters and eigenfunction delocalization.

Contribution

It establishes a probabilistic criterion for eigenfunction equidistribution on tori with random delta potentials, connecting system size, impurity density, and energy levels.

Findings

Eigenfunctions are uniformly distributed with nonzero probability under certain conditions.

The results imply a polynomial lower bound on localization length.

Localization length tends to infinity as energy increases.

Abstract

We study a random Schroedinger operator, the Laplacian with N independently uniformly distributed random delta potentials on flat tori T^d_L = R^d/LZ^d, d = 2, 3, where L > 0 is large. We determine a condition in terms of the size of the torus L, the density of the potentials \rho = N/L^d and the energy of the eigenfunction E such any such eigenfunctions will with nonzero probability be uniformly distributed on the entire torus. We remark that the equidistribution we prove here is still consistent with a localized regime, where the localization length is much larger than the size of the torus. In fact our result implies a certain polynomial lower bound on the localization length, so the localization length becomes infinitely large as E tends to infinity.