Essential self-adjointness, generalized eigenforms, and spectra for the ¯∂-Neumann problem on G-manifolds

Let M be a complex manifold with boundary, satisfying a subelliptic estimate, which is also the total space of a principal G-bundle with G a Lie group and compact orbit space M/G. Here we investigate the ¯∂-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 σ( ) 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. © 2011 Elsevier Inc. All rights reserved.