In this study, we consider the local Cauchy problem for a system of nonlinear Schrödinger equations with non pseudo-conformally invariant interactions in the framework of space of charge and in the framework of space of energy. The main purpose of this study is to construct local solutions in function spaces of analytic vectors for the Galilei generator and the pseudo-conformal generator with data which satisfy exponentially decaying condition at spatial infinity. In particular, we improve the nonlinear estimates have been proved by Hayashi and Kato and Ozawa et al. involving the pseudo-conformal generator with coeﬃcient which depends on time of local existence of solutions and has singularity at finite value.
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