A new proof of long range scattering for critical nonlinear Schr\"odinger equations

TL;DR

This paper introduces a novel proof technique for establishing global existence and long-range scattering in certain nonlinear Schrödinger equations, utilizing Fourier space analysis and stationary phase methods.

Contribution

It provides a new proof approach for long-range scattering in critical nonlinear Schrödinger equations, connecting space-time resonances with phase correction identification.

Findings

Proves global existence for small initial data

Establishes long-range scattering behavior

Identifies phase correction via stationary phase analysis

Abstract

We present a new proof of global existence and long range scattering, from small initial data, for the one-dimensional cubic gauge invariant nonlinear Schr\"odinger equation, and for Hartree equations in dimension $n \geq 2$. The proof relies on an analysis in Fourier space, related to the recent works of Germain, Masmoudi and Shatah on space-time resonances. An interesting feature of our approach is that we are able to identify the long range phase correction term through a very natural stationary phase argument.