Abstract
In this paper we study the homogeneous Wick differential equation associated to the quantum white noise (QWN) Euler operator Delta(g,Q)(E) acting on generalized operators. Delta(g,Q)(E) is defined as sum of the extension of the QWN-Gross Laplacian and the QWN-conservation operator. It is shown that the operator Delta(g,Q)(E) has a representation in terms of the QWN-derivatives {D-c(-), D-c(+) : c is an element of N}. The poisson equation is worked out as a non homogeneous Wick differential equation associated to Delta(g,Q)(E).