Global well-posedness and scattering for the defocusing, $L^{2}$-critical, nonlinear Schr\"odinger equation when $d = 1$

TL;DR

This paper proves global well-posedness and scattering for the defocusing, $L^{2}$-critical nonlinear Schrödinger equation in one dimension, using a frequency localized interaction Morawetz estimate to handle low frequencies.

Contribution

It introduces a frequency localized interaction Morawetz estimate tailored for the $L^{2}$-critical case in one dimension, establishing global results.

Findings

Global well-posedness for the initial value problem

Scattering results for solutions in $L^{2}(R)$

Development of a frequency localized Morawetz estimate

Abstract

In this paper we prove that the defocusing, quintic nonlinear Schr\"odinger initial value problem is globally well-posed and scattering for $u_{0} \in L^{2}(\mathbf{R})$. To do this, we will prove a frequency localized interaction Morawetz estimate similar to the estimate made in [11]. Since we are considering an $L^{2}$ - critical initial value problem we will localize to low frequencies.