The spectrum of continuously perturbed operators and the Laplacian on forms
Date
2019ISSN
0926-2245Source
Differential Geometry and its ApplicationsVolume
65Pages
227-240Google Scholar check
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In this article we study the variation in the spectrum of a self-adjoint nonnegative operator on a Hilbert space under continuous perturbations of the operator. In the particular case of the Laplacian on k-forms over a complete manifold we will use this analytic result to obtain some interesting and significant properties of its spectrum. In particular, we will prove the continuous deformation of the spectrum of the Laplacian under a continuous deformation of the metric of the noncompact manifold. We will also show that the spectrum on 1-forms always contains the function spectrum on any open manifold.