Energy-critical NLS with quadratic potentials

Abstract

We consider the defocusing ${\dot{H}}^1$-critical nonlinear Schrödinger equation in all dimensions ($n\ge 3$) with a quadratic potential $V(x)=\pm \frac{1}{2}|x{|}^2$. We show global well-posedness for radial initial data obeying $\nabla u_0(x),x{u}_0(x)\in L^2$. In view of the potential $V$, this is the natural energy space. In the repulsive case, we also prove scattering. We follow the approach pioneered by Bourgain and Tao in the case of no potential; indeed, we include a proof of their results that incorporates a couple of simplifications discovered while treating the problem with quadratic potential.

Incorporates corrections to Lemma 6.5