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

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