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In this paper, we consider a 3d cubic focusing nonlinear Schr\"odingerequation (NLS) with slowing decaying potentials. Adopting the variationalmethod of Ibrahim-Masmoudi-Nakanishi \cite{IMN}, we obtain a condition forscattering. It is actually sharp in some sense since the solution will blow upif it's false. The proof of blow-up part relies on the method of Du-Wu-Zhang\cite{DWZ}

Dynamics of the focusing 3D cubic NLS with slowly decaying potential