发表机构
Universite Marie \& Louis Pasteur, Laboratoire de Math\'ematiques LmB, UFR Sciences et techniques, 16 route de Gray, 25030 Besan c on CEDEX France
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过傅里叶-维格纳变换将薛定谔-克莱因-戈登系统重写为相空间上的非线性傅里叶-穆瓦亚尔方程，建立适定性与全局解，为汤川耦合提供微局域分析新框架。
AI 中文摘要
我们引入了薛定谔-克莱因-戈登系统的微局域表述，该系统通过汤川耦合描述非相对论量子粒子与克莱因-戈登场之间的相互作用。我们不直接处理薛定谔波函数，而是用其傅里叶-维格纳变换来表示量子分量，并将克莱因-戈登方程改写为复傅里叶变量的形式。消去场变量后，得到一个在相空间上的封闭非线性傅里叶-穆瓦亚尔方程，其未知量是与量子分量相关的傅里叶-维格纳分布。由此得到的表述在微局域层面提供了粒子-场相互作用的精细描述。特别地，原始的三次耦合被转化为由显式双线性算子控制的二次自相互作用。这种新表示允许使用时频分析的技术。在此基础上，我们在包含维纳型范数和指定连续模的各向异性空间中，发展了傅里叶-穆瓦亚尔方程的适定性理论。在紫外截断的一般假设下，我们建立了局部存在性和唯一性，以及微局域正则性的传播。然后，我们利用傅里叶-维格纳变换的紧致性性质，引入了一种新的弱L2解构造，并对准备好的数据获得了全局存在性。其目的是为汤川型相互作用的研究提供一种替代的分析框架，并在非线性色散方程与相空间方法之间建立桥梁。
英文摘要
We introduce a microlocal formulation of the Schr{ö}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{ö}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.
