## 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 $$A_{\mu}(t,x))$$ generated by itself and in an external real potential V(x). x is space variable of the particle, $$x\in {\mathbb{R}}^ d$$. The classical approximation to the quantum field equations of this problem are the Maxwell-Schrödinger equations: $(1)\quad \partial^{\mu}F_{\mu \nu}=J_{\nu};\quad F_{\mu \nu}=\partial_{\mu}A_{\nu}- \partial_{\nu}A_{\mu};\quad (i\partial_ 0+A_ 0)\psi +(\partial_ j-iA_ j)^ 2\psi =V\psi$ 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 $J_ 0=-{\bar \psi}\psi,\quad J_ j=-i(\psi (\partial_ j-iA_ j)\psi -\psi (\overline{\partial_ j-iA_ j)\psi}).$ 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

### MSC:

 35G25 Initial value problems for nonlinear higher-order PDEs 35Q99 Partial differential equations of mathematical physics and other areas of application 78A35 Motion of charged particles 35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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### References:

  DOI: 10.1016/0022-1236(72)90003-1 · Zbl 0229.76018  Tsutsumi M., Mem. Sch. Sci. Eng. Waseda Univ. 43 pp 109– (1979)  Tsutsumi M., Funkcial Ekvac. 23 pp 259– (1980)  DOI: 10.1063/1.524669  DOI: 10.1007/BF02761431 · Zbl 0334.35062  DOI: 10.2977/prims/1195196436 · Zbl 0192.19801  DOI: 10.1007/BF01206943 · Zbl 0486.35048
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