A new proof of scattering below the ground state for the 3d radial focusing cubic NLS

TL;DR

This paper provides a simplified proof of scattering for the 3D radial focusing cubic nonlinear Schrödinger equation below the ground state, avoiding concentration compactness by employing radial Sobolev embedding and virial/Morawetz estimates.

Contribution

It introduces a new, more straightforward proof technique for scattering below the ground state in the 3D radial focusing cubic NLS, bypassing the traditional concentration compactness approach.

Findings

Simplified proof of scattering below the ground state.

Avoidance of concentration compactness method.

Utilization of radial Sobolev embedding and virial/Morawetz estimates.

Abstract

We revisit the scattering result of Holmer and Roudenko on the radial focusing cubic NLS in three space dimensions. Using the radial Sobolev embedding and a virial/Morawetz estimate, we give a simple proof of scattering below the ground state that avoids the use of concentration compactness.