Exponential localization for eigensections of the Bochner-Schrödinger operator
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910428954624000 |
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| author | Kordyukov, Yuri A. |
| author_facet | Kordyukov, Yuri A. |
| contents | We study asymptotic spectral properties of the Bochner-Schrödinger operator $H_{p}=\frac 1pΔ^{L^p\otimes E}+V$ on high tensor powers of a Hermitian line bundle $L$ twisted by a Hermitian vector bundle $E$ on a Riemannian manifold $X$ of bounded geometry under assumption that the curvature form of $L$ is non-degenerate. At an arbitrary point $x_0$ of $X$ the operator $H_p$ can be approximated by a model operator $\mathcal H^{(x_0)}$, which is a Schrödinger operator with constant magnetic field. For large $p$, the spectrum of $H_p$ asymptotically coincides, up to order $p^{-1/4}$, with the union of the spectra of the model operators $\mathcal H^{(x_0)}$ over $X$. We show that, if the union of the spectra of $\mathcal H^{(x_0)}$ over the complement of a compact subset of $X$ has a gap, then the spectrum of $H_{p}$ in the gap is discrete and the corresponding eigensections decay exponentially away the compact subset. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_19684 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Exponential localization for eigensections of the Bochner-Schrödinger operator Kordyukov, Yuri A. Spectral Theory Mathematical Physics Differential Geometry We study asymptotic spectral properties of the Bochner-Schrödinger operator $H_{p}=\frac 1pΔ^{L^p\otimes E}+V$ on high tensor powers of a Hermitian line bundle $L$ twisted by a Hermitian vector bundle $E$ on a Riemannian manifold $X$ of bounded geometry under assumption that the curvature form of $L$ is non-degenerate. At an arbitrary point $x_0$ of $X$ the operator $H_p$ can be approximated by a model operator $\mathcal H^{(x_0)}$, which is a Schrödinger operator with constant magnetic field. For large $p$, the spectrum of $H_p$ asymptotically coincides, up to order $p^{-1/4}$, with the union of the spectra of the model operators $\mathcal H^{(x_0)}$ over $X$. We show that, if the union of the spectra of $\mathcal H^{(x_0)}$ over the complement of a compact subset of $X$ has a gap, then the spectrum of $H_{p}$ in the gap is discrete and the corresponding eigensections decay exponentially away the compact subset. |
| title | Exponential localization for eigensections of the Bochner-Schrödinger operator |
| topic | Spectral Theory Mathematical Physics Differential Geometry |
| url | https://arxiv.org/abs/2404.19684 |