  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.
