Green's Distributions and the Cauchy Problem for the Iterated Klein-Gordon Operator

Abstract

An explicit form of the homogeneous Green's function for the multi-dimensional iterated Klein-Gordon operator is obtained. By a direct calculation from its Fourier representation, the Green's function is expressed as a one-dimensional, infinite integral of the Sonine type. Although this integral is classically divergent when the order of the operator is less than the number of space dimensions, it can be treated rigorously under these conditions using the concepts of distribution analysis. A generalized Sonine integral is developed and the result applied to obtaining an explicit expression for the Green's function, which is now to be regarded as a distribution in the sense of Schwartz. Using a distribution introduced for this purpose, the Green's function is written in a form which explicitly displays its singularities on the light cone. The well-known difference between even- and odd-dimensional spaces is reflected in the nature of these singularities. The singularities appearing for an odd number of space dimensions consist of a finite linear combination of derivatives of the Dirac delta function δ(s2) where s is the space-time distance. The highest derivative appearing is of order ½(n − 2l − 1) with n giving the number of space dimensions and 2l giving the order of the operator. The singular part for even-dimensional spaces consists of a polynomial in 1/s of degree n − 2l + 1. No singularities appear when the order of the operator is greater than the number of dimensions. The general solution of Cauchy's problem for the iterated Klein-Gordon operator is obtained in convolution form. An explicit solution for the ordinary Klein-Gordon equation is presented in a form which exhibits separately the contributions due to the singular part and the regular part of the Green's function.