Eigenvalue distribution in gaps of the essential spectrum of the Bochner-Schrödinger operator
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866918234988478464 |
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| author | Kordyukov, Yuri A. |
| author_facet | Kordyukov, Yuri A. |
| contents | The Bochner-Schrödinger operator $H_{p}=\frac 1pΔ^{L^p}+V$ on high tensor powers $L^p$ of a Hermitian line bundle $L$ on a Riemannian manifold $X$ of bounded geometry is studied 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 of all local Landau levels of the operator at the points of $X$. Moreover, if the union of the local Landau levels over the complement of a compact subset of $X$ has a gap, then the spectrum of $H_{p}$ in the gap is discrete. The main result of the paper is the trace asymptotics formula associated with these eigenvalues. As a consequence, we get a Weyl type asymptotic formula for the eigenvalue counting function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_12928 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Eigenvalue distribution in gaps of the essential spectrum of the Bochner-Schrödinger operator Kordyukov, Yuri A. Spectral Theory Mathematical Physics Differential Geometry The Bochner-Schrödinger operator $H_{p}=\frac 1pΔ^{L^p}+V$ on high tensor powers $L^p$ of a Hermitian line bundle $L$ on a Riemannian manifold $X$ of bounded geometry is studied 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 of all local Landau levels of the operator at the points of $X$. Moreover, if the union of the local Landau levels over the complement of a compact subset of $X$ has a gap, then the spectrum of $H_{p}$ in the gap is discrete. The main result of the paper is the trace asymptotics formula associated with these eigenvalues. As a consequence, we get a Weyl type asymptotic formula for the eigenvalue counting function. |
| title | Eigenvalue distribution in gaps of the essential spectrum of the Bochner-Schrödinger operator |
| topic | Spectral Theory Mathematical Physics Differential Geometry |
| url | https://arxiv.org/abs/2504.12928 |