Solutions to nonlinear Schrodinger equations for special initial data

This article concerns the solvability of the nonlinear Schrodinger equation with gauge invariant power nonlinear term in one space dimension. The well-posedness of this equation is known only for H^s with s >__- 0. Under some assumptions on the nonlinearity, this paper shows that this equation is uniquely solvable for special but typical initial data, namely the linear combinations of 14;(x) and p. v.(1/x), which belong to H^<-1/2-0>. The proof in this article allows L^2-perturbations on the initial data.