  We study the theory of scattering for the Maxwell-Schr"odinger system in
space dimension 3, in the Coulomb gauge. We prove the existence of modified
wave operators for that system with no size restriction on the Schr"odinger and
Maxwell asymptotic 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. The Schr"odinger function is then parametrized in terms of an amplitude
and a phase satisfying a suitable auxiliary system, and the Cauchy problem for
that system, with prescribed asymptotic behaviour determined by the asymptotic
data, is solved by an energy method, thereby leading to solutions of the
original system with prescribed asymptotic behaviour in time. This paper is the
generalization of a previous paper with the same title. However it is entirely
selfcontained and can be read without any previous knowledge of the latter.
