Energy-critical NLS with potentials of quadratic growth

Consider the global wellposedness problem for nonlinear Schr"odinger equation $$i{\mathrm{\partial }}_{t}u=[-\frac{1}{2}\mathrm{\Delta }+V(x)]u\pm \mathrm{\mid }u{\mathrm{\mid }}^{4\mathrm{/}(d-2)}u,u(0)\in \mathrm{\Sigma }({\mathbf{R}}^d),$$ where $\mathrm{\Sigma }$ is the weighted Sobolev space ${\dot{H}}^1\cap \mathrm{\mid }x{\mathrm{\mid }}^{-1}{L}^2$. 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.