Energy-critical NLS with potentials of quadratic growth

2014/11/18 by Jao, Casey · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1411.4950

Abstract

Consider the global wellposedness problem for nonlinear Schrödinger equation i∂_t u = [-\tfrac12 Δ+ V(x)] u ± |u|^4/(d-2) u, u(0) ∈ Σ(R^d), where Σ is the weighted Sobolev space H¹ ∩ |x|^-1 L². The case V(x) = \tfrac12|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.

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