DIFFERENTIAL-OPERATORS FOR SYSTEMS OF LINEAR PARTIAL-DIFFERENTIAL EQUATIONS

The aim of this paper is the representation of solutions of systems of formally hyperbolic differential equations of second order. I. N. VEKUA gave a representation of the solutions using the Riemann-matrix-function. Here we introduce special differential operators which map holomorphic functions into the set of solutions. An existence theorem for such operators is proved which gives a necessary and sufficient condition on the coefficients of a system. These operators are represented explicitly and several properties of them are investigated. We give different representations of the solutions of such systems and discuss the relation between the integral operator method and the method using differential operators which leads to an explicit representation of the Riemann-matrix-function by means of the differential operators. Two examples of special systems with differential operators are given.