Statistical Properties of the one dimensional Anderson model relevant for the Nonlinear Schr\"odinger Equation in a random potential

TL;DR

This paper analyzes the statistical properties of eigenfunction overlaps and eigenenergy combinations in the one-dimensional Anderson model, providing scaling functions that facilitate understanding of the nonlinear Schrödinger equation in random potentials.

Contribution

It introduces the computation of distribution functions for eigenfunction overlaps and eigenenergy combinations, revealing their scaling behavior relevant for nonlinear Schrödinger equation studies.

Findings

Distributions are scaling functions consistent with localization theory.

Distributions enable analysis beyond computational limits.

Results are relevant for understanding noise in nonlinear Schrödinger equations.

Abstract

The statistical properties of overlap sums of groups of four eigenfunctions of the Anderson model for localization as well as combinations of four eigenenergies are computed. Some of the distributions are found to be scaling functions, as expected from the scaling theory for localization. These enable to compute the distributions in regimes that are otherwise beyond the computational resources. These distributions are of great importance for the exploration of the Nonlinear Schr\"odinger Equation (NLSE) in a random potential since in some explorations the terms we study are considered as noise and the present work describes its statistical properties.