2018/01/09 by Charalambous, Nelia, Lu, Zhiqin
#58E30 (Secondary) #58J50 (Primary) #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.1801.02952
In this article we prove a generalization of Weyl's criterion for the spectrum of a self-adjoint nonnegative operator on a Hilbert space. We will apply this new criterion in combination with Cheeger-Fukaya-Gromov and Cheeger-Colding theory to study the k-form essential spectrum over a complete manifold with vanishing curvature at infinity or asymptotically nonnegative Ricci curvature. In addition, we will apply the generalized Weyl criterion to study the variation of the spectrum of a self-adjoint operator under continuous perturbations of the operator. In the particular case of the Laplacian on k-forms over a complete manifold we will use these analytic tools to find significantly stronger results for its spectrum including its behavior under a continuous deformation of the metric of the manifold.