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Scattering for a nonlinear Schrödinger equation with a potential

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  • We consider a 3d cubic focusing nonlinear Schrödinger equation with a potential $$i\partial_t u+\Delta u-Vu+|u|^2u=0,$$ 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 $V=0$ [10, 5]. Moreover, by the concentration-compactness approach, we prove that if $V$ is repulsive, such global solutions scatter.
    Mathematics Subject Classification: 35Q41, 35Q55, 35L15, 35B40, 35B20.

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