Calderón problem for fractional Schrödinger operators on closed Riemannian manifolds
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| Format: | Preprint |
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2024
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| _version_ | 1866911965499097088 |
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| author | Feizmohammadi, Ali Krupchyk, Katya Uhlmann, Gunther |
| author_facet | Feizmohammadi, Ali Krupchyk, Katya Uhlmann, Gunther |
| contents | We study an analog of the anisotropic Calderón problem for fractional Schrödinger operators $(-Δ_g)^α+ V$ with $α\in (0,1)$ on closed Riemannian manifolds of dimensions two and higher. We prove that the knowledge of a Cauchy data set of solutions of the fractional Schrödinger equation, given on an open nonempty a priori known subset of the manifold determines both the Riemannian manifold up to an isometry and the potential up to the corresponding gauge transformation, under certain geometric assumptions on the manifold as well as the observation set. Our method of proof is based on: (i) studying a new variant of the Gel'fand inverse spectral problem without the normalization assumption on the energy of eigenfunctions, and (ii) the discovery of an entanglement principle for nonlocal equations involving two or more compactly supported functions. Our solution to (i) makes connections to antipodal sets as well as local control for eigenfunctions and quantum chaos, while (ii) requires sharp interpolation results for holomorphic functions. We believe that both of these results can find applications in other areas of inverse problems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_16866 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Calderón problem for fractional Schrödinger operators on closed Riemannian manifolds Feizmohammadi, Ali Krupchyk, Katya Uhlmann, Gunther Analysis of PDEs Spectral Theory We study an analog of the anisotropic Calderón problem for fractional Schrödinger operators $(-Δ_g)^α+ V$ with $α\in (0,1)$ on closed Riemannian manifolds of dimensions two and higher. We prove that the knowledge of a Cauchy data set of solutions of the fractional Schrödinger equation, given on an open nonempty a priori known subset of the manifold determines both the Riemannian manifold up to an isometry and the potential up to the corresponding gauge transformation, under certain geometric assumptions on the manifold as well as the observation set. Our method of proof is based on: (i) studying a new variant of the Gel'fand inverse spectral problem without the normalization assumption on the energy of eigenfunctions, and (ii) the discovery of an entanglement principle for nonlocal equations involving two or more compactly supported functions. Our solution to (i) makes connections to antipodal sets as well as local control for eigenfunctions and quantum chaos, while (ii) requires sharp interpolation results for holomorphic functions. We believe that both of these results can find applications in other areas of inverse problems. |
| title | Calderón problem for fractional Schrödinger operators on closed Riemannian manifolds |
| topic | Analysis of PDEs Spectral Theory |
| url | https://arxiv.org/abs/2407.16866 |