Towards a theory of eigenvalue asymptotics on infinite metric graphs: the case of diagonal combs
| Autor: | Kennedy, James B., Mugnolo, Delio, Täufer, Matthias |
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| Rok vydání: | 2024 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 10 pages, 1 figure |
| Databáze: | arXiv |
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