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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/2403.10708 |
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| _version_ | 1866913267101728768 |
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| author | Kennedy, James B. Mugnolo, Delio Täufer, Matthias |
| author_facet | Kennedy, James B. Mugnolo, Delio Täufer, Matthias |
| contents | We examine diagonal combs, a recently identified class of infinite metric graphs whose properties depend on one parameter. These graphs exhibit a fascinating regime where they possess infinite volume while maintaining purely discrete spectrum for the Neumann Laplacian. In this regime, we establish polynomial upper and lower bounds on the $k$-th eigenvalue, revealing that the eigenvalues grow at a rate strictly slower than quadratic. However, once the diagonal combs transition to finite volume, their growth accelerates to a quadratic rate. Our methodology involves employing spectral geometric principles tailored for metric graphs, complemented by deriving estimates for the $k$-th eigenvalue on compact metric graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_10708 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Towards a theory of eigenvalue asymptotics on infinite metric graphs: the case of diagonal combs Kennedy, James B. Mugnolo, Delio Täufer, Matthias Spectral Theory 34B45, 35P15, 81Q35 We examine diagonal combs, a recently identified class of infinite metric graphs whose properties depend on one parameter. These graphs exhibit a fascinating regime where they possess infinite volume while maintaining purely discrete spectrum for the Neumann Laplacian. In this regime, we establish polynomial upper and lower bounds on the $k$-th eigenvalue, revealing that the eigenvalues grow at a rate strictly slower than quadratic. However, once the diagonal combs transition to finite volume, their growth accelerates to a quadratic rate. Our methodology involves employing spectral geometric principles tailored for metric graphs, complemented by deriving estimates for the $k$-th eigenvalue on compact metric graphs. |
| title | Towards a theory of eigenvalue asymptotics on infinite metric graphs: the case of diagonal combs |
| topic | Spectral Theory 34B45, 35P15, 81Q35 |
| url | https://arxiv.org/abs/2403.10708 |