On some spectral properties of the $\overline\partial$-Neumann operator

We consider the $\overline\partial$-Neumann operator
$$ N\colon L^2_{(0,q)}(\Omega ) \longrightarrow L^2_{(0,q)}(\Omega ), $$
where $\Omega \subset \mathbb{C}^n$ is bounded pseudoconvex domain, and
$$N_\varphi\colon L^2_{(0,q)}(\Omega , e^{-\varphi}) \longrightarrow L^2_{(0,q)}(\Omega , e^{-\varphi}),$$
where $\Omega \subseteq \mathbb{C}^n$ is a pseudoconvex domain and $\varphi $ is a plurisubharmonic weight function. $N$ is the inverse to the complex Laplacian $\Box = \overline\partial \, \overline\partial^* + \overline\partial^* \, \overline\partial$.
In addition, we describe spectral properties of the complex Laplacian $\Box_{\varphi,q}$ on weighted spaces $L^2(\mathbb C^n, e^{-\varphi}).$ In this connection it is important to know whether the Fock space
$$\mathcal{A}^2 (\mathbb{C}^n, e^{-\varphi }) =\{ f : \mathbb{C}^n \longrightarrow \mathbb{C} \ {\text{entire}} \ : \int_{\mathbb{C}^n} |f|^2 e^{-\varphi }\, d\lambda < \infty \}$$
is infinite-dimensional, which depends on the behavior at infinity of the eigenvalues of the Levi matrix of the weight function $\varphi$.
We discuss necessary conditions for compactness of the corresponding $\overline\partial$-Neumann operator related to Schrödinger operators with magnetic field.