We present a new proof of global existence and long range scattering, from small initial data, for the one--dimensional cubic gauge invariant nonlinear Schrödinger equation, and for Hartree equations in dimension $n \geq 2$. The proof relies on an analysis in Fourier space, related to the recent works of Germain, Masmoudi, and Shatah on space-time resonances. An interesting feature of our approach is that we are able to identify the long range phase correction term through a very natural stationary phase argument.
"A new proof of long-range scattering for critical nonlinear Schrödinger equations." Differential Integral Equations 24 (9/10) 923 - 940, September/October 2011. https://doi.org/10.57262/die/1356012893