The Schr\"odinger-Klein-Gordon System Revisited

Abstract

We introduce a microlocal formulation of the Schr{\"o}dinger-Klein-Gordon system describing the interaction between a non-relativistic quantum particle and a Klein-Gordon field through Yukawa coupling. Instead of working directly with the Schr{\"o}dinger wave function, we represent the quantum component by its Fourier-Wigner transform, and we rewrite the Klein-Gordon equation in terms of a complex Fourier variable. Eliminating the field variable yields a closed nonlinear Fourier-Moyal equation on phase space whose unknown is the Fourier-Wigner distribution associated with the quantum component. The resulting formulation provides a refined description of the particle-field interaction at the microlocal level. In particular, the original cubic coupling is transformed into a quadratic self-interaction governed by an explicit bilinear operator. This new representation permits the use of techniques from time-frequency analysis. Building on this, we develop a well-posedness theory for the Fourier-Moyal equation in anisotropic spaces involving Wiener-type norms and prescribed moduli of continuity. Local existence and uniqueness are established under general assumptions on the ultraviolet cutoff, together with propagation of microlocal regularity. We then introduce a new construction of weak L 2 -solutions by exploiting compactness properties of Fourier-Wigner transforms and obtain global existence for prepared data. The aim is to provide an alternative analytical framework for the study of Yukawa-type interactions and to establish a bridge between nonlinear dispersive equations and phase-space methods.