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The Cauchy problem for the coupled Maxwell-Schrödinger equations. (English) Zbl 0606.35015

The article deals with a nonrelativistic charged particle with complex scalar field $\psi$ (t,x), moving in the electro-magnetic field (represented in terms of the real vector potential ${A}_{\mu }\left(t,x\right)\right)$ generated by itself and in an external real potential V(x). x is space variable of the particle, $x\in {ℝ}^{d}$. The classical approximation to the quantum field equations of this problem are the Maxwell-Schrödinger equations:

$\left(1\right)\phantom{\rule{1.em}{0ex}}{\partial }^{\mu }{F}_{\mu \nu }={J}_{\nu };\phantom{\rule{1.em}{0ex}}{F}_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu };\phantom{\rule{1.em}{0ex}}\left(i{\partial }_{0}+{A}_{0}\right)\psi +{\left({\partial }_{j}-i{A}_{j}\right)}^{2}\psi =V\psi$

together with the Lorentz gauge condition ${\partial }^{\mu }{A}_{\mu }=0$. ($\mu$,$\nu$ range over 0,1,...,d, whereas j ranges over 1,...,d.) The charge-current densities ${J}_{\nu }$ are

${J}_{0}=-\overline{\psi }\psi ,\phantom{\rule{1.em}{0ex}}{J}_{j}=-i\left(\psi \left({\partial }_{j}-i{A}_{j}\right)\psi -\psi \left(\overline{{\partial }_{j}-i{A}_{j}\right)\psi }\right)·$

The authors consider a Cauchy initial value problem for the system (1) (with initial values fitting to the Problem) and show the existence and uniqueness of a solution on [0,T) for some $T>0$ and any d in a certain function set. If $d=1,2$ one may choose $T=\infty$.

Reviewer: R.Weikard
##### MSC:
 35G25 Initial value problems for nonlinear higher-order PDE 35Q99 PDE of mathematical physics and other areas 78A35 Motion of charged particles 35A05 General existence and uniqueness theorems (PDE) (MSC2000)