Wick differential and Poisson equations associated to the 𝚀𝚆𝙽-Euler operator acting on generalized operators

Abstract

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