The critical case for the concentration of eigenfunctions on singular Riemannian manifolds
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908614946455552 |
|---|---|
| author | Dietze, Charlotte |
| author_facet | Dietze, Charlotte |
| contents | We consider a compact Riemannian manifold with boundary with a certain class of critical singular Riemannian metrics that are singular at the boundary. The corresponding Laplace-Beltrami operator can be seen as a Grushin-type operator plus a potential. We show in the critical case that the average density of eigenfunctions for the Laplace-Beltrami operator with eigenvalues below $λ>0$ is distributed over all length scales between $λ^{-1/2}$ and $1$ near the boundary. We give a precise description of this distribution as $λ\to\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_23520 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The critical case for the concentration of eigenfunctions on singular Riemannian manifolds Dietze, Charlotte Spectral Theory Mathematical Physics Analysis of PDEs 58C40 (Primary) 53C17, 35P20 (Secondary) We consider a compact Riemannian manifold with boundary with a certain class of critical singular Riemannian metrics that are singular at the boundary. The corresponding Laplace-Beltrami operator can be seen as a Grushin-type operator plus a potential. We show in the critical case that the average density of eigenfunctions for the Laplace-Beltrami operator with eigenvalues below $λ>0$ is distributed over all length scales between $λ^{-1/2}$ and $1$ near the boundary. We give a precise description of this distribution as $λ\to\infty$. |
| title | The critical case for the concentration of eigenfunctions on singular Riemannian manifolds |
| topic | Spectral Theory Mathematical Physics Analysis of PDEs 58C40 (Primary) 53C17, 35P20 (Secondary) |
| url | https://arxiv.org/abs/2510.23520 |