Semiclassical trace formula for the Bochner-Schrödinger operator
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917926136709120 |
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| author | Kordyukov, Yuri A. |
| author_facet | Kordyukov, Yuri A. |
| contents | We study the semiclassical Bochner-Schrödinger operator $H_{p}=\frac{1}{p^2}Δ^{L^p\otimes E}+V$ on tensor powers $L^p$ of a Hermitian line bundle $L$ twisted by a Hermitian vector bundle $E$ on a Riemannian manifold of bounded geometry. For any function $φ\in C^\infty_c(\mathbb R)$, we consider the bounded linear operator $φ(H_p)$ in $L^2(X,L^p\otimes E)$ defined by the spectral theorem. We prove that its smooth Schwartz kernel on the diagonal admits a complete asymptotic expansion in powers of $p^{-1}$ in the semiclassical limit $p\to \infty$. In particular, when the manifold is compact, we get a complete asymptotic expansion for the trace of $φ(H_p)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_12087 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Semiclassical trace formula for the Bochner-Schrödinger operator Kordyukov, Yuri A. Differential Geometry Mathematical Physics Spectral Theory We study the semiclassical Bochner-Schrödinger operator $H_{p}=\frac{1}{p^2}Δ^{L^p\otimes E}+V$ on tensor powers $L^p$ of a Hermitian line bundle $L$ twisted by a Hermitian vector bundle $E$ on a Riemannian manifold of bounded geometry. For any function $φ\in C^\infty_c(\mathbb R)$, we consider the bounded linear operator $φ(H_p)$ in $L^2(X,L^p\otimes E)$ defined by the spectral theorem. We prove that its smooth Schwartz kernel on the diagonal admits a complete asymptotic expansion in powers of $p^{-1}$ in the semiclassical limit $p\to \infty$. In particular, when the manifold is compact, we get a complete asymptotic expansion for the trace of $φ(H_p)$. |
| title | Semiclassical trace formula for the Bochner-Schrödinger operator |
| topic | Differential Geometry Mathematical Physics Spectral Theory |
| url | https://arxiv.org/abs/2502.12087 |