Stability of Gel'fand's inverse interior spectral problem for Schrödinger operators
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913951095193600 |
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| author | Lu, Jinpeng |
| author_facet | Lu, Jinpeng |
| contents | We study Gel'fand's inverse interior spectral problem of determining a closed Riemannian manifold $(M,g)$ and a potential function $q$ from the knowledge of the eigenvalues $λ_j$ of the Schrödinger operator $-Δ_g + q$ and the restriction of the eigenfunctions $ϕ_j|_U$ on a given open subset $U\subset M$, where $Δ_g$ is the Laplace-Beltrami operator on $(M,g)$. We prove that an approximation of finitely many spectral data on $U$ determines a finite metric space that is close to $(M,g)$ in the Gromov-Hausdorff topology, and further determines a discrete function that approximates the potential $q$ with uniform estimates. This leads to a quantitative stability estimate for the inverse interior spectral problem for Schrödinger operators in the general case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_15560 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stability of Gel'fand's inverse interior spectral problem for Schrödinger operators Lu, Jinpeng Analysis of PDEs Differential Geometry 35R30, 58J50, 53B21 We study Gel'fand's inverse interior spectral problem of determining a closed Riemannian manifold $(M,g)$ and a potential function $q$ from the knowledge of the eigenvalues $λ_j$ of the Schrödinger operator $-Δ_g + q$ and the restriction of the eigenfunctions $ϕ_j|_U$ on a given open subset $U\subset M$, where $Δ_g$ is the Laplace-Beltrami operator on $(M,g)$. We prove that an approximation of finitely many spectral data on $U$ determines a finite metric space that is close to $(M,g)$ in the Gromov-Hausdorff topology, and further determines a discrete function that approximates the potential $q$ with uniform estimates. This leads to a quantitative stability estimate for the inverse interior spectral problem for Schrödinger operators in the general case. |
| title | Stability of Gel'fand's inverse interior spectral problem for Schrödinger operators |
| topic | Analysis of PDEs Differential Geometry 35R30, 58J50, 53B21 |
| url | https://arxiv.org/abs/2507.15560 |