On the distribution of linear combinations of eigenvalues of the Anderson model

TL;DR

This paper derives probabilistic estimates for linear combinations of eigenvalues in the 1D Anderson model, which are crucial for understanding perturbative aspects of the nonlinear Schrödinger equation.

Contribution

It provides new probabilistic bounds on linear combinations of eigenvalues, extending beyond previous density estimates by Wegner and Minami.

Findings

Probabilistic estimates for linear combinations of eigenvalues.

Extension of eigenvalue density estimates to linear combinations.

Relevance to nonlinear Schrödinger equation perturbation analysis.

Abstract

Probabilistic estimates on linear combinations of eigenvalues of the one dimensional Anderson model are derived. So far only estimates on the density of eigenvalues and of pairs were found by Wegner and by Minami. Our work was motivated by perturbative explorations of the Nonlinear Schroedinger Equation, where linear combinations of eigenvalues are the denominators and evaluation of their smallness is crucial.