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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 $A\sb{\mu}(t,x))$ generated by itself and in an external real potential V(x). x is space variable of the particle, $x\in {\bbfR}\sp d$. The classical approximation to the quantum field equations of this problem are the Maxwell-Schrödinger equations: $$ (1)\quad \partial\sp{\mu}F\sb{\mu \nu}=J\sb{\nu};\quad F\sb{\mu \nu}=\partial\sb{\mu}A\sb{\nu}- \partial\sb{\nu}A\sb{\mu};\quad (i\partial\sb 0+A\sb 0)\psi +(\partial\sb j-iA\sb j)\sp 2\psi =V\psi$$ together with the Lorentz gauge condition $\partial\sp{\mu}A\sb{\mu}=0$. ($\mu$,$\nu$ range over 0,1,...,d, whereas j ranges over 1,...,d.) The charge-current densities $J\sb{\nu}$ are $$ J\sb 0=-{\bar \psi}\psi,\quad J\sb j=-i(\psi (\partial\sb j-iA\sb j)\psi -\psi (\overline{\partial\sb j-iA\sb j)\psi}). $$ 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

35G25Initial value problems for nonlinear higher-order PDE
35Q99PDE of mathematical physics and other areas
78A35Motion of charged particles
35A05General existence and uniqueness theorems (PDE) (MSC2000)
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