Wave and inverse wave operators for the quadratic nonlinear Schrödinger equations in 3D

Abstract

Our purpose in this paper is to prove existence of wave operator $\mathcal{W}_{+}$ or inverse wave operator $\widetilde{\mathcal{W}}_{+}$ for nonlinear Schrödinger equations with quadratic nonlinearities in three space dimensions. Our results show that the mapping $\mathcal{W}_{+} \widetilde{\mathcal{W}}_{+}$ is well defined and are improvement of results on the range of inverse wave operator obtained in [6].