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We study the theory of scattering for the Maxwell-Schr"odinger system inspace dimension 3,in the Coulomb gauge.In the special case of vanishingasymptotic magnetic field,we prove the existence of modified wave operators forthat system with no size restriction on the Schr"odinger data and we determinethe asymptotic behaviour in time of solutions in the range of the waveoperators.The method consists in partially solving the Maxwell equations forthe potentials,substituting the result into the Schr"odinger equation,whichthen becomes both nonlinear and nonlocal in time,and treating the latter by themethod previously used for the Hartree equation and for the Wave-Schr"odingersystem.

Long Range Scattering and Modified Wave Operators for the Maxwell-Schr�dinger System I. The Case of Vanishing Asymptotic Magnetic Field