Abstract:

In this paper, we consider the solution of the equation ◇k(p±i0)=∑mi=0Cr ◇rδ, where ◇k is introduced and named as the Diamond operator iterated k-times and is defined by ◇=[(∂2/∂x21+... +∂2/∂x2p)2-(∂2/∂x2p+1+... +∂2/∂x2p+q)2]k Let x = (x1, x2, ..., xn) be a point of the n-dimensional Euclidean space. Consider a non-degenerate quadratic form in n variables of the form P = P(x) = x12+...+xp2- xp+12 - ... - xp+q2, where p + q = n, Cr is a constant, δ is the delta distribution ◇0δ = δ and k = 0, 1, .... The distributions (P ± i0)λ are defined by (P±i0)λ = limε→0{P±iε|x|2}λ where ε > 0, |x|2 = x12 + ... + xn2, λ εC. The distributions (P ± i0)λ are an important contribution of Gelfand (cf. [1], p. 274). The distributions (P ± i0)λ are analytic in λ everywhere except at λ = -n/2 - k, k = 0, 1, ..., where they have simple poles (cf. [1], p. 275). By causal (anticausal) distributions, we mean distributions where P = P(x) = x12 + ... + xn-12 - xn2. The causal distributions are particularly important when n = 4 because they appear frequently in the quantum theory of field. In this note we obtain the solutions of the causal and anticausal n-dimensional Diamond operator by following, line by line, the paper entitled "On the solutions of the n-dimensional Diamond operator" by Amnuay Kananthai (cf. [2]).