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

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

2011-01-10

arxiv.org/abs/1101.1863v1

J. Funct. Anal., v261 (2011) 2717-2740

Mathematics

Spectral Theory

25 pages

Scientific paper

10.1016/j.jfa.2011.07.010

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.

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