On Eigenvalues of Logarithmic Potential Operator in the Hyperbolic Space
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911404313804800 |
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| author | Johnson, Jiya Rose Verma, Sheela |
| author_facet | Johnson, Jiya Rose Verma, Sheela |
| contents | Let $Ω$ be a bounded open set in the Poincaré hyperbolic disk, $\mathbb{D}$. In this article, we consider the hyperbolic logarithmic potential operator $\mathcal{L}_h : L^2(Ω) \to L^2(Ω)$, defined by \begin{equation*}
\mathcal{L}_h u(z)=\frac{1}{2}\int_Ω\log\frac{1}{[z,w]}\,u(w)\, {\,\rm d}(w), \end{equation*} and the associated eigenvalue problem on $Ω$ \begin{equation}
\mathcal{L}_h u=τu. \end{equation} We first extend the notion of polarization with respect to hyperplanes in the Poincaré disk and prove the associated properties. Then we establish a reverse Faber-Krahn inequality for the largest eigenvalue, $τ_{h}$ of $\mathcal{L}_h$, under polarization. Further, we provide a representation formula for the eigenfunctions of $\mathcal{L}_h$. In addition, we show that the operator $\mathcal{L}_h$ is a positive operator on $L^2(Ω)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_20431 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Eigenvalues of Logarithmic Potential Operator in the Hyperbolic Space Johnson, Jiya Rose Verma, Sheela Analysis of PDEs Let $Ω$ be a bounded open set in the Poincaré hyperbolic disk, $\mathbb{D}$. In this article, we consider the hyperbolic logarithmic potential operator $\mathcal{L}_h : L^2(Ω) \to L^2(Ω)$, defined by \begin{equation*} \mathcal{L}_h u(z)=\frac{1}{2}\int_Ω\log\frac{1}{[z,w]}\,u(w)\, {\,\rm d}(w), \end{equation*} and the associated eigenvalue problem on $Ω$ \begin{equation} \mathcal{L}_h u=τu. \end{equation} We first extend the notion of polarization with respect to hyperplanes in the Poincaré disk and prove the associated properties. Then we establish a reverse Faber-Krahn inequality for the largest eigenvalue, $τ_{h}$ of $\mathcal{L}_h$, under polarization. Further, we provide a representation formula for the eigenfunctions of $\mathcal{L}_h$. In addition, we show that the operator $\mathcal{L}_h$ is a positive operator on $L^2(Ω)$. |
| title | On Eigenvalues of Logarithmic Potential Operator in the Hyperbolic Space |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2601.20431 |