Quantified asymptotic analysis for the relativistic quantum mechanical system with electromagnetic fields

Abstract

We study the asymptotic analysis of a relativistic quantum mechanical system interacting with self-consistent electromagnetic fields, with a specific focus on the Maxwell–Klein–Gordon (MKG) model. Firstly, we show that the MKG system is globally well-posed under the Coulomb gauge condition, even in the presence of self-interacting potential. Furthermore, to derive the asymptotic analysis, we consider the non-relativistic and semi-classical regime simultaneously, introducing a single scaling parameter that serves to parameterize both the speed of light and the Planck constant. As the scaling parameter approaches zero, we obtain rigorous and quantified estimates on the asymptotic convergence of the electrostatic MKG system toward the classical Euler–Poisson system. Our analytical framework relies on the modulated energy estimate.