Global well-posedness of NLS with a rough potential below the energy norm

TL;DR

This paper proves global well-posedness for a 3D cubic defocusing nonlinear Schrödinger equation with a rough potential in Sobolev spaces below the energy norm, using a spectral theory-enhanced I-method.

Contribution

It extends the I-method to handle rough external potentials in NLS, establishing well-posedness in lower regularity spaces than previously known.

Findings

Global well-posedness in H^s for s>5/6

Extension of harmonic analysis tools via spectral theory

Adaptation of the I-method for rough potentials

Abstract

We show that in the presence of a rough external potential, a 3d cubic defocusing NLS is globally well-posed in H^s for s>5/6. The proof is based on the approach of Colliander-Keel-Staffilani-Takaoka-Tao, called the I-method, but in order to deal with a rough potential, we modify harmonic analysis tools by spectral theory.