Global well-posedness and scattering for the defocusing, $L^{2}$-critical, nonlinear Schr{\"o}dinger equation when $d \geq 3$

TL;DR

This paper proves that the defocusing, mass-critical nonlinear Schrödinger equation in dimensions three and higher is globally well-posed and exhibits scattering for initial data in L^2, using a frequency localized interaction Morawetz estimate.

Contribution

It introduces a frequency localized interaction Morawetz estimate to establish global well-posedness and scattering for the L^2-critical nonlinear Schrödinger equation in dimensions d ≥ 3.

Findings

Global well-posedness for initial data in L^2

Scattering behavior established in high dimensions

Development of a frequency localized Morawetz estimate

Abstract

In this paper we prove that the defocusing, $d$-dimensional mass critical nonlinear Schr{\"o}dinger initial value problem is globally well-posed and scattering for $u_{0} \in L^{2}(\mathbf{R}^{d})$ and $d \geq 3$. To do this, we will prove a frequency localized interaction Morawetz estimate similar to the estimate made in [10]. Since we are considering an $L^{2}$ - critical initial value problem we will localize to low frequencies.