Global well-posedness and scattering for the mass critical nonlinear Schr{\"o}dinger equation with mass below the mass of the ground state

In this paper we prove that the focusing, $d$-dimensional mass critical nonlinear Schr{"o}dinger initial value problem is globally well-posed and scattering for $u_0\in L^2({\mathbf{R}}^d)$, $| u_{0} |{L^{2}($\mathbf{R}$^{d})} < | Q |{L^{2}($\mathbf{R}$^{d})}$,where$Q$ is the ground state, and $d\ge 1$. We first establish an interaction Morawetz estimate that is positive definite when $| u_{0} |{L^{2}($\mathbf{R}$^{d})} < | Q |{L^{2}($\mathbf{R}$^{d})}$, and has the appropriate scaling. Next, we will prove a frequency localized interaction Morawetz estimate similar to the estimates made in \cite{D2}, \cite{D3}, \cite{D4}. See also \cite{CKSTT4} for the energy critical case. Since we are considering an $L^2$ - critical initial value problem we will localize to low frequencies.