Energy-Critical NLS with Quadratic Potentials

We consider the defocusing -critical nonlinear Schrödinger equation in all dimensions (n ≥ 3) with a quadratic potential . We show global well-posedness for radial initial data obeying ∇u 0(x), xu 0(x) ∊ 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.