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

We derive a necessary condition for compactness of the weighted $\overline\partial$-Neumann operator on the space $L^2(\mathbb C^n,e^{-\varphi})$, under the assumption that the corresponding weighted Bergman space of entire functions has infinite dimension. Moreover, we compute the essential spectrum of the complex Laplacian for decoupled weights, $\varphi(z) = \varphi_1(z_1) + \dotsb + \varphi_n(z_n)$, and investigate (non-) compactness of the $\overline\partial$-Neumann operator in this case. More can be said if every $\Delta\varphi_j$ defines a nontrivial doubling measure.