Determinantal point processes associated with the Bochner-Schrödinger operator
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914563419537408 |
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| author | Kordyukov, Yuri A. |
| author_facet | Kordyukov, Yuri A. |
| contents | We consider the Bochner-Schrödinger operator $H_{p}=\frac 1pΔ^{L^p}+V$ on tensor powers $L^p$ of a Hermitian line bundle $L$ on a Riemannian manifold $X$ of bounded geometry under the assumption of non-degeneracy of the curvature form of $L$. For large $p$, the spectrum of $H_p$ asymptotically coincides with the union $Σ$ of all local Landau levels of the operator at the points of $X$. We study the determinantal point process on $X$ associated with the spectral projection of $H_p$ corresponding to an interval $I=(α,β)$ such that $α,β\not \in Σ$ and compute the asymptotics of its linear statistics as $p$ goes to infinity. When $X$ is compact, this implies the law of large numbers and central limit theorem for the corresponding empirical measures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_13575 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Determinantal point processes associated with the Bochner-Schrödinger operator Kordyukov, Yuri A. Differential Geometry Mathematical Physics Probability Spectral Theory We consider the Bochner-Schrödinger operator $H_{p}=\frac 1pΔ^{L^p}+V$ on tensor powers $L^p$ of a Hermitian line bundle $L$ on a Riemannian manifold $X$ of bounded geometry under the assumption of non-degeneracy of the curvature form of $L$. For large $p$, the spectrum of $H_p$ asymptotically coincides with the union $Σ$ of all local Landau levels of the operator at the points of $X$. We study the determinantal point process on $X$ associated with the spectral projection of $H_p$ corresponding to an interval $I=(α,β)$ such that $α,β\not \in Σ$ and compute the asymptotics of its linear statistics as $p$ goes to infinity. When $X$ is compact, this implies the law of large numbers and central limit theorem for the corresponding empirical measures. |
| title | Determinantal point processes associated with the Bochner-Schrödinger operator |
| topic | Differential Geometry Mathematical Physics Probability Spectral Theory |
| url | https://arxiv.org/abs/2605.13575 |