The Energy-Critical Quantum Harmonic Oscillator

We consider the energy critical nonlinear Schr"{o}dinger equation in dimensions $d\ge 3$ with a harmonic oscillator potential $V(x) = \tfrac{1}{2} |x|^2$. When the nonlinearity is defocusing, we prove global wellposedness for all initial data in the energy space $\mathrm{\Sigma }$, consisting of all functions $u_0$ such that both $\mathrm{\nabla }{u}_0$ and $x{u}_0$ belong to $L^2$. This result extends a theorem of Killip-Visan-Zhang \cite{kvz_quadratic_potentials}, which treats the radial case. For the focusing problem, we obtain global wellposedness for all data satisfying an analogue of the usual size restriction in terms of the ground state $W$. The proof uses the concentration compactness variant of the induction on energy paradigm. In particular, we develop a linear profile decomposition adapted to the propagator $\exp[ it(\tfrac{1}{2}\Delta - \tfrac{1}{2}|x|^2)]$forboundedsequencesin$\Sigma$.