Energy-critical NLS with potentials of quadratic growth

Abstract

Consider the global wellposedness problem for nonlinear Schrödinger equation \[ i\partial_t u = [-\tfrac{1}{2} Δ+ V(x)] u \pm |u|^{4/(d-2)} u, \ u(0) \in Σ(\mathbf{R}^d), \] where $\Sigma$ is the weighted Sobolev space ${\dot{H}}^1\cap |x{|}^{-1}{L}^2$. The case $V(x)=\frac{1}{2}|x{|}^2$ 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.

arXiv commentSome expository improvements. arXiv admin note: text overlap with arXiv:1406.2289