Anisotropic Calderón problem for a logarithmic Schrödinger operator of order $2+$ on closed Riemannian manifolds
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866918185956016128 |
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| author | Das, Saumyajit Ghosh, Tuhin Pramanik, Susovan |
| author_facet | Das, Saumyajit Ghosh, Tuhin Pramanik, Susovan |
| contents | In this article, we study the anisotropic Calderón problems for the non local logarithimic Schrödinger operators $(-Δ_g+m)\log{(-Δ_g+m)}+V$ with $m>1$ on a closed, connected, smooth Riemannian manifold of dimension $n\geq2$. We will show that, for the operator $(-Δ_g+m)\log{(-Δ_g+m)}+V$, the recovery of both the Riemannian metric and the potential is possible from the Cauchy data, in the setting of a common underlying manifold with varying metrics. This result is unconditional. The last result can be extended to the case of setwise distinct manifolds also. In particular, we demonstrate that for setwise distinct manifolds, the Cauchy data associated with the operator $(-Δ_g+m)\log{(-Δ_g+m)}+V$, measured on a suitable non-empty open subset, uniquely determines the Riemannian manifold up to isometry and the potential up to an appropriate gauge transformation. This particular result is unconditional when the potential is supported entirely within the observation set. In the more general setting-where the potential may take nonzero values outside the observation set-specific geometric assumptions are required on both the observation set and the unknown region of the manifold. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_02409 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Anisotropic Calderón problem for a logarithmic Schrödinger operator of order $2+$ on closed Riemannian manifolds Das, Saumyajit Ghosh, Tuhin Pramanik, Susovan Analysis of PDEs 35S05, 58J35, 58J40 In this article, we study the anisotropic Calderón problems for the non local logarithimic Schrödinger operators $(-Δ_g+m)\log{(-Δ_g+m)}+V$ with $m>1$ on a closed, connected, smooth Riemannian manifold of dimension $n\geq2$. We will show that, for the operator $(-Δ_g+m)\log{(-Δ_g+m)}+V$, the recovery of both the Riemannian metric and the potential is possible from the Cauchy data, in the setting of a common underlying manifold with varying metrics. This result is unconditional. The last result can be extended to the case of setwise distinct manifolds also. In particular, we demonstrate that for setwise distinct manifolds, the Cauchy data associated with the operator $(-Δ_g+m)\log{(-Δ_g+m)}+V$, measured on a suitable non-empty open subset, uniquely determines the Riemannian manifold up to isometry and the potential up to an appropriate gauge transformation. This particular result is unconditional when the potential is supported entirely within the observation set. In the more general setting-where the potential may take nonzero values outside the observation set-specific geometric assumptions are required on both the observation set and the unknown region of the manifold. |
| title | Anisotropic Calderón problem for a logarithmic Schrödinger operator of order $2+$ on closed Riemannian manifolds |
| topic | Analysis of PDEs 35S05, 58J35, 58J40 |
| url | https://arxiv.org/abs/2511.02409 |