Essential self-adjointness, generalized eigenforms, and spectra for the [formula omitted]-Neumann problem on G-manifolds

Abstract

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.