Optimal eigenvalues on a metric graph with densities
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918261380087808 |
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| author | Naderi, Kiyan Nicolussi, Noema |
| author_facet | Naderi, Kiyan Nicolussi, Noema |
| contents | We introduce and study Laplacians on a finite metric graph endowed with generalized densities, that is, measures of finite mass. One important motivation is that this setting provides a common framework for several interesting classes of operators: discrete graph Laplacians, Kirchhoff Laplacians and Dirichlet-to-Neumann operators on graphs. Our main interest lies in spectral optimization with respect to the underlying measure. In contrast to the setting of domains and manifolds, we prove that a minimal $k$-th eigenvalue exists, whereas the corresponding maximization problem has no meaning. We then establish connections between these optimal eigenvalues and the geometry of the metric graph, including a transparent geometric characterization of the first optimal eigenvalue via the resistance metric, and a Weyl law for the higher optimal eigenvalues. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_20508 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal eigenvalues on a metric graph with densities Naderi, Kiyan Nicolussi, Noema Spectral Theory Mathematical Physics 81Q35, 35R02, 35P15, 35P20 We introduce and study Laplacians on a finite metric graph endowed with generalized densities, that is, measures of finite mass. One important motivation is that this setting provides a common framework for several interesting classes of operators: discrete graph Laplacians, Kirchhoff Laplacians and Dirichlet-to-Neumann operators on graphs. Our main interest lies in spectral optimization with respect to the underlying measure. In contrast to the setting of domains and manifolds, we prove that a minimal $k$-th eigenvalue exists, whereas the corresponding maximization problem has no meaning. We then establish connections between these optimal eigenvalues and the geometry of the metric graph, including a transparent geometric characterization of the first optimal eigenvalue via the resistance metric, and a Weyl law for the higher optimal eigenvalues. |
| title | Optimal eigenvalues on a metric graph with densities |
| topic | Spectral Theory Mathematical Physics 81Q35, 35R02, 35P15, 35P20 |
| url | https://arxiv.org/abs/2512.20508 |