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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2401.02136 |
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| _version_ | 1866929197598900224 |
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| author | Charalambous, Nelia Lu, Zhiqin |
| author_facet | Charalambous, Nelia Lu, Zhiqin |
| contents | In this article, we find sufficient conditions on an open Riemannian manifold so that a Weyl criterion holds for the $L^p$-spectrum of the Laplacian on $k$-forms, and also prove the decomposition of the $L^p$-spectrum depending on the order of the forms. We then show that the resolvent set of an operator such as the Laplacian on $L^p$ lies outside a parabola whenever the volume of the manifold has an exponential volume growth rate, removing the requirement on the manifold to be of bounded geometry. We conclude by providing a detailed description of the $L^p$ spectrum of the Laplacian on $k$-forms over hyperbolic space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_02136 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $L^p$-spectral theory for the Laplacian on forms Charalambous, Nelia Lu, Zhiqin Differential Geometry Analysis of PDEs Spectral Theory 58J50 In this article, we find sufficient conditions on an open Riemannian manifold so that a Weyl criterion holds for the $L^p$-spectrum of the Laplacian on $k$-forms, and also prove the decomposition of the $L^p$-spectrum depending on the order of the forms. We then show that the resolvent set of an operator such as the Laplacian on $L^p$ lies outside a parabola whenever the volume of the manifold has an exponential volume growth rate, removing the requirement on the manifold to be of bounded geometry. We conclude by providing a detailed description of the $L^p$ spectrum of the Laplacian on $k$-forms over hyperbolic space. |
| title | $L^p$-spectral theory for the Laplacian on forms |
| topic | Differential Geometry Analysis of PDEs Spectral Theory 58J50 |
| url | https://arxiv.org/abs/2401.02136 |