Energy-critical NLS with quadratic potentials

We consider the defocusing H˙ 1-critical nonlinear Schr¨odinger equation in all dimensions (n ≥ 3) with a quadratic potential V (x) = ± 1/2|x|^2. We show global well-posedness for radial initial data obeying ∇u0(x), xu0(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.