Long Range Scattering and Modified Wave Operators for the Wave-Schr"odinger System III

TL;DR

The paper extends long-range scattering theory for the Wave-Schrödinger system in 3D to more regular Schrödinger data (no Fourier support restriction) by a time-inversion and amplitude-phase reformulation. An auxiliary system for the amplitude and phase is analyzed, with a long/short range split enabling the construction of modified wave operators that map asymptotic data $(u_+,A_+,\dot A_+)$ to a global solution $(u,A)$. It provides a rigorous local construction near zero for the auxiliary system, builds the corresponding wave operators, and proves quantitative asymptotics toward explicit free-wave+nonlinear corrections $(u_a,A_a)$ with precise decay rates. These results generalize prior work to higher regularity and strengthen the framework toward more complex coupled systems such as Maxwell–Schrödinger.

Abstract

We continue the study of scattering theory for the system consisting of a Schr"odinger equation and a wave equation with a Yukawa type coupling in space dimension 3. In previous papers, we proved the existence of modified wave operators for that system with no size restriction on the data and we determined the asymptotic behaviour in time of the solutions in the range of the wave operators, first under a support condition on the Schr"odinger asymptotic state and then without that condition, but for solutions of relatively low regularity. Here we extend the latter result to the case of more regular solutions.