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

TL;DR

This paper investigates spectral properties and compactness criteria of the weighted ar-Neumann operator on complex spaces, providing conditions related to the weight function and its Laplacian, with explicit spectrum calculations for certain weights.

Contribution

It establishes a necessary condition for compactness of the weighted ar-Neumann operator and computes its essential spectrum for decoupled weights, advancing understanding of spectral behavior in weighted complex analysis.

Findings

Derived a necessary condition for compactness of the operator

Computed the essential spectrum for decoupled weights

Analyzed the impact of eltaar Laplacian measures on spectral properties

Abstract

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.