The spectrum of the Laplacian on forms
Abstract
In this article we prove a generalization of Weyl's criterion for the spectrum of a selfadjoint nonnegative operator on a Hilbert space. We will apply this new criterion in combination with CheegerFukayaGromov and CheegerColding 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 selfadjoint 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.
 Publication:

arXiv eprints
 Pub Date:
 January 2018
 arXiv:
 arXiv:1801.02952
 Bibcode:
 2018arXiv180102952C
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Analysis of PDEs;
 Mathematics  Spectral Theory;
 58J50 (Primary);
 58E30 (Secondary)