# The spectrum of the Laplacian on forms.

@article{Charalambous2016TheSO, title={The spectrum of the Laplacian on forms.}, author={Nelia Charalambous and Zhiqin Lu}, journal={arXiv: Differential Geometry}, year={2016} }

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…

## 2 Citations

The spectrum of continuously perturbed operators and the Laplacian on forms

- MathematicsDifferential Geometry and its Applications
- 2019

The spectrum of the Laplacian on forms over flat manifolds

- MathematicsMathematische Zeitschrift
- 2019

In this article we prove that the spectrum of the Laplacian on $k$-forms over a noncompact flat manifold is always a connected closed interval of the nonnegative real line. The proof is based on a…

## References

SHOWING 1-10 OF 33 REFERENCES

On the Lp independence of the spectrum of the Hodge Laplacian on non-compact manifolds

- Mathematics
- 2005

The spectrum of the Laplacian on a manifold of nonnegative Ricci curvature

- Mathematics
- 1997

The study of the spectrum of the Laplacian on a complete noncompact Riemannian manifold has received much attention during the past decade or so. In particular, it has been conjectured and partially…

On the Lp-Spectrum of Uniformly Elliptic Operators on Riemannian Manifolds

- Mathematics
- 1993

Abstract We prove that the L p spectrum of uniformly elliptic divergence form operators on a complete Riemannian manifold is independent of p ∈ [1, ∞] if the volume of the manifold grows uniformly…

On the spectrum of the Laplacian

- Mathematics
- 2014

In this article we prove a generalization of Weyl’s criterion for the essential spectrum of a self-adjoint operator on a Hilbert space. We then apply this criterion to the Laplacian on functions over…

Eigenvalues of the Laplacian on forms

- Mathematics
- 1982

Some bounds for eigenvalues of the Laplace operator acting on forms on a compact Riemannian manifold are derived. In case of manifolds without boundary we give upper bounds in terms of the curvature,…

ON THE UPPER ESTIMATE OF THE HEAT KERNEL OF A COMPLETE RIEMANNIAN MANIFOLD

- Mathematics
- 1981

Let M be a complete non-compact Riemannian manifold whose sectional curvature is bounded between two constants -k and K. Then one expects that the heat diffusion in such a manifold behaves like the…