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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*G*_{N} (*x*) and *G*_{N} ^{1}(*x*), defined in the usual way by independent paths of integration in the *k* _{0} 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 *G*_{N} (*x*; odd *n*) contains a finite jump on the light cone as well as a linear combination of derivatives up to order ½(*n* − 2*N* − 1) of δ(*x* ^{2}); the singular part of *G*_{N} ^{1}(*x*; odd *n*) consists of a logarithmic singularity ln (|*x* ^{2}|) along with a polynomial in (*x* ^{2})^{−1} of degree ½(*n* − 2*N* − 1). For even‐dimensional spaces, the singular part of both Green's functions consists of a polynomial in (*x* ^{2})^{−1/2} of degree *n* − 2*N* + 1 vanishing *outside* the light cone for *G*_{N} and vanishing *inside* the light cone for *G*_{N} ^{1}. In all cases no singularities or finite jumps occur when the order 2*N* 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.

© 1962 The American Institute of Physics

Received 09 April 1962
Published online 22 December 2004