The $\partial $-complex on the Segal–Bargmann space
We study certain densely defined unbounded operators on the Segal–Bargmann space. These are the annihilation and creation operators of quantum mechanics. In several complex variables we have the $\partial $-operator and its adjoint $\partial ^*$ acting on $(p,0)$-forms with coefficients in the Segal–Bargmann space. We consider the corresponding $\partial $-complex and study the spectral properties of the corresponding complex Laplacian $\tilde\Box = \partial \partial ^* + \partial ^*\partial $. Finally, we study a more general complex Laplacian $\tilde\Box _D = D D^* + D^* D$, where $D$ is a differential operator of polynomial type, to find the canonical solutions to the inhomogeneous equations $Du=\alpha $ and $D^*v=\beta $.