Threshold solutions for the focusing 3D cubic Schrodinger equation

We study the focusing 3d cubic NLS equation with H-1 data at the mass-energy threshold, namely, when M[u(0)]E[u(0)] = M[Q]E[Q]. In earlier works of Holmer-Roudenko and Duyckaerts-Holmer-Roudenko, the behavior of solutions (i.e., scattering and blow up in finite time) was classified when M[u(0)][Eu-0] < M[Q]E[Q]. In this paper, we first exhibit 3 special solutions: e(it)Q and Q(+/-), where Q is the ground state, Q(+/-) exponentially approach the ground state solution in the positive time direction, Q(+) has finite time blow up and Q(-) scatters in the negative time direction. Secondly, we classify solutions at this threshold and obtain that up to (H) over dot1/2 symmetries, they behave exactly as the above three special solutions, or scatter and blow up in both time directions as the solutions below the mass-energy threshold. These results are obtained by studying the spectral properties of the linearized Schrodinger operator in this mass-supercritical case, establishing relevant modulational stability and careful analysis of the exponentially decaying solutions to the linearized equation.