Green's Distributions Associated with the Operator [□m − (−μ2)m]l

Abstract

The results of a previous paper on homogeneous Green's functions for a multi-dimensional iterated Klein-Gordon operator (□ + μ2)l are extended to include homogeneous Green's functions associated with the operator [□m (− μ2)m]l in multi-dimensional spaces. The Fourier representation of the Green's functions may be expressed, after some angular integrations, as a one-dimensional infinite integral which does not in general converge. Using the concepts of distribution analysis, it is shown how this improper integral can be evaluated directly to get explicit expressions for the Green's functions. The Green's functions themselves must then be interpreted as distributions in the sense of Schwartz. Several distributions instrumental in this treatment are introduced and their properties studied. Explicit expressions for the singularities of the Green's functions on the light cone are presented. The well-known difference between even- and odd-dimensional spaces is reflected in the nature of these singularities. The singularities appearing for odd-dimensional spaces 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−2m−l) with n giving the number of space dimensions. The singular part for even-dimensional spaces consists of a polynomial in 1/s of degree n − 2ml + 1. No singularities appear when the order of the operator is greater than the number of space dimensions. Finally, a complete set of homogeneous Δ-function solutions is given along with their initial conditions at zero time. All of these functions would be needed in obtaining the general solution to the Cauchy problem for the operator considered.