Principal spectral rigidity implies subprincipal spectral rigidity
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912293372035072 |
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| author | de Hoop, Maarten V. Ilmavirta, Joonas Katsnelson, Vitaly |
| author_facet | de Hoop, Maarten V. Ilmavirta, Joonas Katsnelson, Vitaly |
| contents | We study the inverse spectral problem of jointly recovering a radially symmetric Riemannian metric and an additional coefficient from the Dirichlet spectrum of a perturbed Laplace-Beltrami operator on a bounded domain. Specifically, we consider the elliptic operator \[ L_{a,b} := e^{a-b} \nabla \cdot e^b \nabla \] on the unit ball $ B \subset \mathbb{R}^3 $, where the scalar functions $ a = a(|x|) $ and $ b = b(|x|) $ are spherically symmetric and satisfy certain geometric conditions. While the function $ a $ influences the principal symbol of $ L $, the function $ b $ appears in its first-order terms. We investigate the extent to which the Dirichlet eigenvalues of $ L_{a,b} $ uniquely determine the pair $ (a, b) $ and establish spectral rigidity results under suitable assumptions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_19866 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Principal spectral rigidity implies subprincipal spectral rigidity de Hoop, Maarten V. Ilmavirta, Joonas Katsnelson, Vitaly Analysis of PDEs We study the inverse spectral problem of jointly recovering a radially symmetric Riemannian metric and an additional coefficient from the Dirichlet spectrum of a perturbed Laplace-Beltrami operator on a bounded domain. Specifically, we consider the elliptic operator \[ L_{a,b} := e^{a-b} \nabla \cdot e^b \nabla \] on the unit ball $ B \subset \mathbb{R}^3 $, where the scalar functions $ a = a(|x|) $ and $ b = b(|x|) $ are spherically symmetric and satisfy certain geometric conditions. While the function $ a $ influences the principal symbol of $ L $, the function $ b $ appears in its first-order terms. We investigate the extent to which the Dirichlet eigenvalues of $ L_{a,b} $ uniquely determine the pair $ (a, b) $ and establish spectral rigidity results under suitable assumptions. |
| title | Principal spectral rigidity implies subprincipal spectral rigidity |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2503.19866 |