The energy-critical quantum harmonic oscillator

We consider the energy critical nonlinear Schrödinger equation in dimensions 3 and above with a harmonic oscillator potential. In the defocusing situation, we prove global wellposedness for all initial data in the energy space Σ. This extends a result of Killip-Visan-Zhang, who treated the radial case. For the focusing nonlinearity, we obtain wellposedness for data in Σ satisfying an analogue of the usual size restriction in terms of the ground state W. We implement the concentration compactness variant of the induction on energy paradigm and, in particular, develop profile decompositions adapted to the harmonic oscillator.