Long Range Scattering and Modified Wave Operators for the Maxwell-Schr"odinger system I.The case of vanishing asymptotic magnetic field

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