Scattering for a Nonlinear Schrödinger Equation with a Potential

We consider a 3d cubic focusing nonlinear Schrödinger equation with a potential $i\partial_t u+Δu-Vu+|u|^2u=0,$ 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.

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