Analytic smoothing effect for global solutions to a quadratic system of nonlinear Schrödinger equations

We consider the Cauchy problem for a system of nonlinear Schrödinger equations in the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-subcritical setting or in the scale critical Sobolev setting. In particular, we study analytic smoothing effect in space variables for a system of nonlinear Schrödinger equations under the mass resonance condition by applying the method has been studied in Sasaki (J Funct Anal 270:1064–1090, 2016) and the mass conservation law has been proved in Hayashi et al. (Ann Inst Henri Poincaré-AN 30:661–690, 2013), with large data which satisfy exponentially decaying condition at spatial infinity. Also we discus analytic smoothing effect in space variables in the scale critical Sobolev setting with data which have sufficiently small norm and satisfy exponentially decaying condition at spatial infinity.