A semilinear Schroedinger equation with random potential

TL;DR

This paper investigates a semilinear Schrödinger equation with a localized, random, time-dependent potential, demonstrating that solutions remain bounded and scatter on average, supported by sharp dispersive estimates.

Contribution

It provides new dispersive estimates for Schrödinger equations with random potentials and shows energy boundedness and scattering in both linear and nonlinear cases.

Findings

Solutions scatter on average

Energy remains bounded on average

Dispersive estimates are established for random potentials

Abstract

We study a non-linear Schroedinger equation with a Hartree-type nonlinearity and a localized random time-dependent external potential. Sharp dispersive estimates for the linear Schroedinger equation with a random time-dependent potential enable us to also treat the case of small semi-linear perturbations. In both the linear and the nonlinear instances, we prove that, on average, energy remains bounded and solutions scatter.