A new proof of long range scattering for critical nonlinear Schrödinger equations
Jun Kato, Fabio Pusateri
TL;DR
The paper develops a new Fourier-space, space-time resonance–based method to prove global existence and long-range scattering for small data solutions of the one-dimensional cubic gauge-invariant NLS andHartree equations in $n\ge 2$. The core idea is a stationary-phase analysis of the Fourier-space Duhamel integral for the profile, which reveals a natural phase correction $- i \int \frac{1}{s} |\hat f(s,\xi)|^2 \hat f(s,\xi) ds$ and an appropriately decaying remainder, yielding explicit modified wave operators. The authors derive sharp $L^\infty$ decay bounds $\|u(t)\|_{L^\infty} \lesssim \varepsilon (1+|t|)^{-1/2}$ in 1D and $\|u(t)\|_{L^\infty} \lesssim \varepsilon (1+|t|)^{-n/2}$ for Hartree, together with asymptotic formulas involving a phase $\frac{|x|^2}{4t}$ and a log-term, describing the modified scattering behavior. This framework avoids vector-field techniques and clarifies the origin of the phase correction, suggesting applicability to other critical scattering problems via space-time resonance methods.
Abstract
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.