Scattering for a Nonlinear Schr\"odinger Equation with a Potential

We consider a 3d cubic focusing nonlinear Schr"odinger equation with a potential $$i{\mathrm{\partial }}_{t}u+\mathrm{\Delta }u-Vu+\mathrm{\mid }u{\mathrm{\mid }}^{2}u=0,$$ where $V$ is a real-valued short-range potential having a small negative part. We find criteria for global well-posedness analogous to the homogeneous case $V=0$ (Duyckaerts-Holmer-Roudenko). Moreover, by the concentration-compactness approach, we prove that if $V$ is repulsive, such global solutions scatter.