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

Nakamitsu, Kuniaki; Tsutsumi, Masayoshi

J. Math. Phys. 27, 211-216 (1986).

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)

Keywords:

nonrelativistic charged particle; complex scalar field; electro-magnetic field; quantum field equations; Maxwell-Schrödinger equations; Lorentz gauge condition; Cauchy initial value problem; existence; uniqueness

References:

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