Long range scattering for the Wave-Schr\"odinger system revisited

We reconsider the theory of scattering for the Wave-Schr"odinger system and more precisely the local Cauchy problem with infinite initial time, which is the main step in the construction of the wave operators. Using a method due to Nakanishi, we eliminate a loss of regularity between the Schr"odinger asymptotic data and the Schr"odinger solution in the treatment of that problem, in the special case of vanishing asymptotic data for the wave field.