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

Abstract

We derive a necessary condition for compactness of the weighted $\overline{\partial }$-Neumann operator on the space $L^2({\mathbb{C}}^n,e^{-\phi })$, 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, $\phi (z)={\phi }_1(z_1)+\cdots +{\phi }_n(z_n)$, and investigate (non-) compactness of the $\overline{\partial }$-Neumann operator in this case. More can be said if every $\Delta {\phi }_j$ defines a nontrivial doubling measure.

11 pages; fixed some mistakes and added new results

References in corpus (2)

• Pointwise estimates for the Bergman kernel of the weighted Fock space
• Essential spectra of tensor product Hilbert complexes, and the $\overline{\partial }$-Neumann problem on product manifolds