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Abstract

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 $.