Remark on the scattering operator for the quintic nonlinear Dirac equation in one space dimension

Abstract

This paper is concerned with the scattering operator $S$ for the one dimensional Dirac equation with a quintic nonlinearity. It has been proved that $S$ can be defined on a neighborhood of 0 in the Sobolev space $H^\kappa (\mathbb{R};\mathbb{C}^2)$ for any $\kappa > 3/4$. In the present paper, we prove that for any $M \in \mathbb{N}$ and $s \ge \max\{ \kappa,M \}$, there exists some neighborhood $U$ of 0 in the weighted Sobolev space $H^{s,M}(\mathbb{R};\mathbb{C}^2)$ such that $S(U) \subset H^{s,M}(\mathbb{R};\mathbb{C}^2)$.