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

Abstract

Explicit forms of the Green's functions (which are to be regarded as distributions in the sense of Schwartz) for the multi-mass Klein-Gordon operator in n-dimensional spaces are presented. The homogeneous Green's functions GN(x) and GN1(x), defined in the usual way by independent paths of integration in the k0 plane, are investigated in the neighborhood of the light cone. The parameter N indicates the total number of masses involved. The singularities on the light cone reflect the well-known difference between even- and odd-dimensional wave propagation. It is found that GN(x; odd n) contains a finite jump on the light cone as well as a linear combination of derivatives up to order ½(n − 2N − 1) of δ(x2); the singular part of GN1(x; odd n) consists of a logarithmic singularity ln (|x2|) along with a polynomial in (x2)−1 of degree ½(n − 2N − 1). For even-dimensional spaces, the singular part of both Green's functions consists of a polynomial in (x2)−1/2 of degree n − 2N + 1 vanishing outside the light cone for GN and vanishing inside the light cone for GN1. In all cases no singularities or finite jumps occur when the order 2N of the operator is greater than the number n + 1 of space-time dimensions. The general solution of the Cauchy problem is given both for the data carrying surface t = 0 and for arbitrary spacelike data surfaces.