The $\partial$-complex on the Fock space

Abstract

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 ${\partial }^{\ast }$ 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 $\tilde{\square }=\partial {\partial }^{\ast }+{\partial }^{\ast }\partial .$ Finally we study a more general complex Laplacian ${\tilde{\square }}_D=D{D}^{\ast }+D^{\ast }D,$ where $D$ is a differential operator of polynomial type, to find the canonical solutions to the inhomogeneous equations $Du=\alpha$ and $D^{\ast }v=\beta .$

22 pages