Scattering for the non-radial 3D cubic nonlinear Schroedinger equation

Scattering of radial $H^1$ solutions to the 3D focusing cubic nonlinear Schr"odinger equation below a mass-energy threshold $M[u]E[u]<M[Q]E[Q]$ and satisfying an initial mass-gradient bound $|u_0|{L^2} |\nabla u_0 |{L^2} < |Q|{L^2} |\nabla Q|{L^2}$,where$Q$ is the ground state, was established in Holmer-Roudenko (2007). In this note, we extend the result in Holmer-Roudenko (2007) to non-radial $H^1$ data. For this, we prove a non-radial profile decomposition involving a spatial translation parameter. Then, in the spirit of Kenig-Merle (2006), we control via momentum conservation the rate of divergence of the spatial translation parameter and by a convexity argument based on a local virial identity deduce scattering. An application to the defocusing case is also mentioned.