Weyl Law for Schrödinger Operators on Noncompact Manifolds, Heat Kernel, and Karamata-Hardy-Littlewood Theorem
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913992898772992 |
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| author | Dai, Xianzhe Yan, Junrong |
| author_facet | Dai, Xianzhe Yan, Junrong |
| contents | Building on our earlier work on heat kernel asymptotics for Schrödinger-type operators on noncompact manifolds, we establish both the classical and semiclassical Weyl laws for Schrödinger operators of the form $Δ+V$ and $\hbar^2Δ+V$ on complete noncompact manifolds. While the semiclassical law can be approached via localization, the classical Weyl law has remained widely expected but unproven in this generality. We impose a mild bounded integral oscillation condition on $ V $ in addition to the assumptions that $V$ diverges at infinity and satisfies a doubling condition. In this setting, our oscillation condition is sharp and strictly weaker than all previously known assumptions, even in the Euclidean case.
A central novelty of our approach is an extended Karamata-Hardy-Littlewood Tauberian theorem, adapted to accommodate non-regularly varying spectral asymptotics in noncompact settings, together with its semiclassical analogue. These Tauberian tools allow us to derive both versions of Weyl's law within a unified framework. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_15551 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weyl Law for Schrödinger Operators on Noncompact Manifolds, Heat Kernel, and Karamata-Hardy-Littlewood Theorem Dai, Xianzhe Yan, Junrong Differential Geometry Analysis of PDEs Spectral Theory Building on our earlier work on heat kernel asymptotics for Schrödinger-type operators on noncompact manifolds, we establish both the classical and semiclassical Weyl laws for Schrödinger operators of the form $Δ+V$ and $\hbar^2Δ+V$ on complete noncompact manifolds. While the semiclassical law can be approached via localization, the classical Weyl law has remained widely expected but unproven in this generality. We impose a mild bounded integral oscillation condition on $ V $ in addition to the assumptions that $V$ diverges at infinity and satisfies a doubling condition. In this setting, our oscillation condition is sharp and strictly weaker than all previously known assumptions, even in the Euclidean case. A central novelty of our approach is an extended Karamata-Hardy-Littlewood Tauberian theorem, adapted to accommodate non-regularly varying spectral asymptotics in noncompact settings, together with its semiclassical analogue. These Tauberian tools allow us to derive both versions of Weyl's law within a unified framework. |
| title | Weyl Law for Schrödinger Operators on Noncompact Manifolds, Heat Kernel, and Karamata-Hardy-Littlewood Theorem |
| topic | Differential Geometry Analysis of PDEs Spectral Theory |
| url | https://arxiv.org/abs/2504.15551 |