On the Solutions of the Causal And Anticausaln -Dimensional Diamond Operator

In this paper, we consider the solution of the equation \diamondsuit^{k}(P\pm i0) = \sum_{r=0}^{m} C_{r}\diamondsuit^{r}\delta where \diamondsuit^{k} is introduced and named as the Diamond operator iterated k -times and is defined by \diamondsuit^{k} = \left[\left({\partial^{2} \over \partial x_{1}^{2}}+ \cdots + {\partial^{2} \over \partial x_{p}^{2}}\right)^{2} - \left({\partial^{2} \over \partial x_{p+1}^{2}} + \cdots + {\partial^{2} \over \partial x_{p+q}^{2}}\right)^{2}\right]^{k}. Let x = (x_{1},x_{2},\ldots,x_{n}) be a point of the n -dimensional Euclidean space. Consider a non-degenerate quadratic form in n variables of the form P = P(x) = x_{1}^{2} + \cdots + x_{p}^{2} - x_{p+1}^{2} - \cdots - x_{p+q}^{2} , where p + q = n , C_{r} is a constant, \delta is the delta distribution \diamondsuit^{0}\delta =\delta and k = 0,1,\ldots. The distributions (P\pm i0)^{\lambda} are defined by (P\pm i0)^{\lambda} = \lim_{\varepsilon \rightarrow 0}\ \{P\pm i\varepsilon \vert x\vert^{2}\}^{\lambda} , where \vare...