The generalized $\mathrm{\partial }$-complex on the Segal Bargmann space

We study certain densely defined unbounded operators on the Segal-Barg-mann space, related to the annihilation and creation operators of quantum mechanics. We consider the corresponding $D$-complex and study properties of the corresponding complex Laplacian ${\tilde{\mathrm{\square }}}_D=D{D}^{\ast }+D^{\ast }D,$ where $D$ is a differential operator of polynomial type.