  We study certain densely defined unbounded operators on the Fock 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 Fock space. We consider the
corresponding $\partial$-complex and study 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 $D^*v=\beta.$
