Global well-posedness and scattering for the mass critical nonlinear Schrodinger equation with mass below the mass of the ground state

In this paper we prove that the focusing, d-dimensional mass critical nonlinear Schrodinger initial value problem is globally well-posed and scattering for u(0) is an element of L-2(R-d), parallel to u(0)parallel to(L2(Rd)) < parallel to Q parallel to(L2(Rd)), where Q is the ground state, and d >= 1. We first establish an interaction Morawetz estimate that is positive definite when parallel to u(0)parallel to(L2(Rd)) < parallel to Q parallel to(L2(Rd)), and has the appropriate scaling. Next, we will prove an interaction Morawetz estimate localized to low frequencies, similar to the estimates made in [13-15]. See also [12] for localization to high frequencies in the energy critical case. (C) 2015 Published by Elsevier Inc.