Discreteness of spectrum for the ∂-Neumann Laplacian on manifolds of bounded geometry

2018/08/08 by Berger, Franz
#32W05 #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.1808.02730

Abstract

For a Hermitian holomorphic vector bundle over a Hermitian manifold, we consider the Dolbeault Laplacian with ∂-Neumann boundary conditions, which is a self-adjoint operator on the space of square-integrable differential forms with values in the given holomorphic bundle. We argue that some known results on the spectral properties of this operator on pseudoconvex domains in \mathbb Cⁿ continue to hold on Kähler manifolds satisfying certain bounded geometry assumptions. In particular, we will consider the Dolbeault complex for forms with values in a line bundle, where known results from magnetic Schrödinger operator theory can be applied.

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