Dynamics near the origin of the long range scattering for the one-dimensional Schrödinger equation (Q6644101)

In this paper, the author addresses the nonlinear scattering problem for the 1D cubic NLS equation on the real line. It is well-known that the asymptotic behavior of a small solution of the 1D cubic NLS involves a logarithmic phase correction, and that the scattering theory requires modifications. The aim of this work is to provide a detailed description of the modified wave operators constructed by \textit{T. Ozawa} [Commun. Math. Phys. 139, No. 3, 479--493 (1991; Zbl 0742.35043)] and \textit{N. Hayashi} and \textit{P. I. Naumkin} [Am. J. Math. 120, No. 2, 369--389 (1998; Zbl 0917.35128)], respectively.