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

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.