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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 ψ (t,x), moving in the electro-magnetic field (represented in terms of the real vector potential A μ (t,x)) generated by itself and in an external real potential V(x). x is space variable of the particle, x d . The classical approximation to the quantum field equations of this problem are the Maxwell-Schrödinger equations:

(1) μ F μν =J ν ;F μν = μ A ν - ν A μ ;(i 0 +A 0 )ψ+( j -iA j ) 2 ψ=Vψ

together with the Lorentz gauge condition μ A μ =0. (μ,ν range over 0,1,...,d, whereas j ranges over 1,...,d.) The charge-current densities J ν are

J 0 =-ψ ¯ψ,J j =-i(ψ( j -iA j )ψ-ψ( j -iA j )ψ ¯)·

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=.

Reviewer: R.Weikard
MSC:
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)