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

TL;DR

This paper revisits the scattering theory for the Wave-Schr"odinger system, focusing on the local Cauchy problem at infinite initial time, and improves the regularity treatment using Nakanishi's method.

Contribution

It introduces a refined approach to the local Cauchy problem for the Wave-Schr"odinger system, eliminating regularity loss in the case of zero asymptotic wave data.

Findings

Elimination of regularity loss in the scattering analysis

Application of Nakanishi's method to the Wave-Schr"odinger system

Enhanced understanding of the local Cauchy problem at infinite initial time

Abstract

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.