We discuss Donsker's delta function within the framework of White Noise Analysis, in particular its extension to complex arguments. With a view towards applications to quantum physics we also study sums and products of Donsker's delta functions.
Abstract In this paper we study the homogeneous Wick differential equation associated to the quantum white noise (𝚀𝚆𝙽) Euler operator Δ E g , Q ΔEg,Q acting on generalized operators. Δ E g , Q ΔEg,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 ΔEg,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 ΔEg,Q .