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We prove sharp L^ decay and modified scattering for the Hartree nonlinear Schr\"odinger equation in dimensions 2 and 3 using the testing by wavepackets method of Ifrim and Tataru. We show that the scattering behavior happens at a regularity well below that of earlier results of Hayashi-Naumkin and Kato-Pusateri.
Tim Van Hoose (Thu,) studied this question.