Dispersive Estimates for Nonlinear Schr\"odinger Equations with External Potentials

TL;DR

This paper establishes optimal decay estimates for nonlinear Schrödinger equations with external potentials, demonstrating that solutions decay similarly to free solutions under small initial data in higher dimensions.

Contribution

It provides the first optimal decay estimates for Hartree type nonlinear Schrödinger equations with external potentials in three or more dimensions.

Findings

Proved decay estimates comparable to free solutions.

Controlled high Sobolev norms to estimate solution behavior.

Applicable to small initial data in higher dimensions.

Abstract

We consider the long time dynamics of nonlinear Schr\"odinger equations with an external potential. More precisely, we look at Hartree type equations in three or higher dimensions with small initial data. We prove an optimal decay estimate, which is comparable to the decay of free solutions. Our proof relies on good control on a high Sobolev norm of the solution to estimate the terms in Duhamel's formula.