Energy-critical NLS with potentials of quadratic growth

TL;DR

This paper extends the analysis of the energy-critical nonlinear Schrödinger equation to a broader class of potentials that grow quadratically, using microlocal analysis techniques to handle the lack of explicit propagator formulas.

Contribution

It generalizes previous results from harmonic oscillator potentials to approximately quadratic potentials using advanced microlocal analysis methods.

Findings

Successfully extended well-posedness results to new potential classes.

Applied Fourier integral parametrix techniques in nonlinear PDE analysis.

Maintained concentration compactness framework for broader potentials.

Abstract

Consider the global wellposedness problem for nonlinear Schr\"odinger equation \[ i\partial_t u = [-\tfrac{1}{2} \Delta + V(x)] u \pm |u|^{4/(d-2)} u, \ u(0) \in \Sigma(\mathbf{R}^d), \] where $\Sigma$ is the weighted Sobolev space $\dot{H}^1 \cap |x|^{-1} L^2$. The case $V(x) = \tfrac{1}{2}|x|^2$ was recently treated by the author. This note generalizes the results to a class of "approximately quadratic" potentials. We closely follow the previous concentration compactness arguments for the harmonic oscillator. A key technical difference is that in the absence of a concrete formula for the linear propagator, we apply more general tools from microlocal analysis, including a Fourier integral parametrix of Fujiwara.