The Energy-Critical Quantum Harmonic Oscillator

Abstract

We consider the energy critical nonlinear Schrödinger equation in dimensions $d\ge 3$ with a harmonic oscillator potential $V(x)=\frac{1}{2}|x{|}^2$. When the nonlinearity is defocusing, we prove global wellposedness for all initial data in the energy space $\Sigma$, consisting of all functions $u_0$ such that both $\nabla u_0$ 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 $\exp [it(\frac{1}{2}\Delta -\frac{1}{2}|x{|}^2)]$ for bounded sequences in $\Sigma$.

Corrected typos

References in corpus (1)

• The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions