On nonlinear Schr\"odinger equations with random potentials: existence and probabilistic properties

TL;DR

This paper investigates nonlinear Schrödinger equations with random potentials, establishing existence of solutions almost surely and analyzing their probabilistic properties, including limit theorems and bounds on solution norms.

Contribution

It provides new conditions for solution existence and probabilistic analysis for both continuum and discrete random potentials in nonlinear Schrödinger equations.

Findings

Solutions exist almost surely under certain potential conditions

Probabilistic properties like CLT and LLN are established for solutions

Expected bounds on the $L^{ abla}$-norm of solutions depending on potential size

Abstract

In this paper we are concerned with nonlinear Schr\"odinger equations with random potentials. Our class includes continuum and discrete potentials. Conditions on the potential $V_{\omega}$ are found for existence of solutions almost sure $\omega $. We study probabilistic properties like central limit theorem and law of larger numbers for the obtained solutions by independent ensembles. We also give estimates on the expected value for the $L^{\infty}$-norm of the solution showing how it depends on the size of the potential.