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In this article we study the asymptotic behavior of a quadratic NLS equationwith small, time-dependent potential and small spatially localized initialdata. We prove global existence and scattering of solutions. The two mainingredients of the proof are the space-time resonance method and theboundedness of wave operators for the linear Schr\"{o}dinger equation withpotential.

Global existence and scattering for quadratic NLS with potential in 3D