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

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.

25 pages

References in corpus (5)

• Essential self-adjointness for semi-bounded magnetic Schrödinger operators on non-compact manifolds
• Generalized eigenfunctions and spectral theory for strongly local Dirichlet forms
• The Allegretto-Piepenbrink Theorem for strongly local forms
• The Levi Problem On Strongly Pseudoconvex $G$-Bundles
• Heat kernel estimates for the $\bar{\partial }$-Neumann problem on $G$-manifolds

Cited by in corpus (2)

• Heat kernel estimates for the $\bar{\partial }$-Neumann problem on $G$-manifolds
• Bergman spaces of natural G-manifolds