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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/2411.18874 |
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| _version_ | 1866910738833997824 |
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| author | Xie, Bing Zhao, Yigeng Zhao, Yongqiang |
| author_facet | Xie, Bing Zhao, Yigeng Zhao, Yongqiang |
| contents | It is well known that the standard flat torus $\mathbb{T}^2=\mathbb{R}^2/\Z^2$ has arbitrarily large Laplacian-eigenvalue multiplicities. We prove, however, that $24$ is the optimal upper bound for the multiplicities of the nonzero eigenvalues of a $2$-dimensional discrete torus. For general higher dimension discrete tori, we characterize the eigenvalues with large multiplicities. As consequences, we get uniform boundedness results of the multiplicity for a long range and an optimal global bound for the multiplicity. Our main tool of proof is the theory of vanishing sums of roots of unity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_18874 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the multiplicity of the eigenvalues of discrete tori Xie, Bing Zhao, Yigeng Zhao, Yongqiang Spectral Theory It is well known that the standard flat torus $\mathbb{T}^2=\mathbb{R}^2/\Z^2$ has arbitrarily large Laplacian-eigenvalue multiplicities. We prove, however, that $24$ is the optimal upper bound for the multiplicities of the nonzero eigenvalues of a $2$-dimensional discrete torus. For general higher dimension discrete tori, we characterize the eigenvalues with large multiplicities. As consequences, we get uniform boundedness results of the multiplicity for a long range and an optimal global bound for the multiplicity. Our main tool of proof is the theory of vanishing sums of roots of unity. |
| title | On the multiplicity of the eigenvalues of discrete tori |
| topic | Spectral Theory |
| url | https://arxiv.org/abs/2411.18874 |