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Scattering of radial $H^1$ solutions to the 3D focusing cubic nonlinearSchr\"odinger equation below a mass-energy threshold $M[u]E[u] < M[Q]E[Q]$ andsatisfying 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, wasestablished in Holmer-Roudenko (2007). In this note, we extend the result inHolmer-Roudenko (2007) to non-radial $H^1$ data. For this, we prove anon-radial profile decomposition involving a spatial translation parameter.Then, in the spirit of Kenig-Merle (2006), we control via momentum conservationthe rate of divergence of the spatial translation parameter and by a convexityargument based on a local virial identity deduce scattering. An application tothe defocusing case is also mentioned.

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