The Energy-Critical Quantum Harmonic Oscillator

TL;DR

This paper proves global well-posedness for the energy-critical nonlinear Schrödinger equation with a harmonic oscillator potential in higher dimensions, extending previous radial results and employing concentration compactness and profile decompositions.

Contribution

It extends global well-posedness results to non-radial initial data and develops a linear profile decomposition for the harmonic oscillator propagator.

Findings

Global well-posedness for defocusing case in energy space

Global solutions for focusing case under ground state size restrictions

Development of a linear profile decomposition for the harmonic oscillator propagator

Abstract

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 $\Sigma$, 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 $\exp[ it(\tfrac{1}{2}\Delta - \tfrac{1}{2}|x|^2)]$ for bounded sequences in $\Sigma$.