The article deals with a nonrelativistic charged particle with complex scalar field (t,x), moving in the electro-magnetic field (represented in terms of the real vector potential generated by itself and in an external real potential V(x). x is space variable of the particle, . The classical approximation to the quantum field equations of this problem are the Maxwell-Schrödinger equations:
together with the Lorentz gauge condition . (, range over 0,1,...,d, whereas j ranges over 1,...,d.) The charge-current densities are
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 and any d in a certain function set. If one may choose .