The $\partial$-complex on the Fock space

TL;DR

This paper investigates the spectral properties of the $ar{ ext{partial}}$-complex and associated Laplacians on the Fock space, extending to polynomial-type operators and providing solutions to related inhomogeneous equations.

Contribution

It introduces a spectral analysis of the $ar{ ext{partial}}$-complex on the Fock space and generalizes to polynomial-type operators for solving inhomogeneous equations.

Findings

Spectral properties of the complex Laplacian $ ilde ox$ are characterized.

Extension to a more general Laplacian $ ilde ox_D$ for polynomial-type operators.

Canonical solutions to inhomogeneous equations involving $D$ and $D^*$ are obtained.

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 $\partial^*$ 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 $D^*v=\beta.$