Essential self-adjointness, generalized eigenforms, and spectra for the $\bar\partial$-Neumann problem on $G$-manifolds

TL;DR

This paper studies the spectral properties of the $ard$-Neumann Laplacian on certain complex manifolds with group actions, establishing essential self-adjointness and linking spectrum to generalized eigenforms.

Contribution

It proves the essential self-adjointness of the $ard$-Neumann Laplacian on $G$-manifolds and connects spectrum characterization to the existence of generalized eigenforms.

Findings

Laplacian is essentially self-adjoint on compactly supported forms.

Spectrum corresponds to energies with subexponentially bounded generalized eigenforms.

Almost all energies in the spectrum have associated well-behaved eigenforms.

Abstract

Let $M$ be a strongly pseudoconvex complex manifold which is also the total space of a principal $G$-bundle with $G$ a Lie group and compact orbit space $\bar M/G$. Here we investigate the $\bar\partial$-Neumann Laplacian on $M$. We show that it is essentially self-adjoint on its restriction to compactly supported smooth forms. Moreover we relate its spectrum to the existence of generalized eigenforms: an energy belongs to $\sigma(\square)$ if there is a subexponentially bounded generalized eigenform for this energy. Vice versa, there is an expansion in terms of these well-behaved eigenforms so that, spectrally, almost every energy comes with such a generalized eigenform.