Threshold solutions for the focusing 3D cubic Schrödinger 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) is classified when M[u_0]E[u_0] < M[Q]E[Q]. In this paper, we first exhibit 3 special solutions: e^{it}Q and Q^+, Q^-; here Q is the ground state, and Q^+, Q^- exponentially approach the ground state solution in the positive time direction, meanwhile Q^+ having finite time blow up and Q^- scattering in the negative time direction. Secondly, we classify solutions at this threshold and obtain that up to \dot{H}^{1/2} 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 Schroedinger operator in this mass-supercritical case, establishing relevant modulational stability and careful analysis of the exponentially decaying solutions to the linearized equation.