A sharp condition for scattering of the radial 3D cubic nonlinear Schrodinger equation

We consider the problem of identifying sharp criteria under which radial H-1 (finite energy) solutions to the focusing 3d cubic nonlinear Schrodinger equation (NLS) i partial derivative(t)u + Delta u + vertical bar u vertical bar(2)u = 0 scatter, i.e., approach the solution to a linear Schrodinger equation as t -> +/-infinity. The criteria is expressed in terms of the scale-invariant quantities vertical bar vertical bar u(0 vertical bar vertical bar)L(2 vertical bar vertical bar) del u(0 vertical bar vertical bar)L(2) and M[u] E[u], where u(0) denotes the initial data, and M[u] and E[u] denote the ( conserved in time) mass and energy of the corresponding solution u( t). The focusing NLS possesses a soliton solution e(it) Q(x), where Q is the ground-state solution to a nonlinear elliptic equation, and we prove that if M[u] E[u] < M[Q] E[Q] and vertical bar vertical bar u(0 vertical bar vertical bar) L-2 vertical bar vertical bar del u(0 vertical bar vertical bar) L-2 < vertical bar vertical bar Q vertical bar vertical bar L-2 vertical bar vertical bar del Q vertical bar vertical bar L-2, then the solution u(t) is globally well-posed and scatters. This condition is sharp in the sense that the soliton solution e(it) Q(x), for which equality in these conditions is obtained, is global but does not scatter. We further show that if M[u] E[u] < M[Q] E[Q] and vertical bar vertical bar u(0 vertical bar vertical bar) L-2 vertical bar vertical bar del u(0) vertical bar vertical bar L-2 > vertical bar vertical bar Q vertical bar vertical bar L-2 vertical bar vertical bar del Q vertical bar vertical bar L-2, then the solution blows-up in finite time. The technique employed is parallel to that employed by Kenig-Merle [17] in their study of the energy-critical NLS.