Improved semiclassical eigenvalue estimates for the Laplacian and the Landau Hamiltonian
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| Format: | Preprint |
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2025
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| _version_ | 1866911305604005888 |
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| author | Frank, Rupert L. Larson, Simon Pfeiffer, Paul |
| author_facet | Frank, Rupert L. Larson, Simon Pfeiffer, Paul |
| contents | The Berezin--Li--Yau and the Kröger inequalities show that Riesz means of order $\geq 1$ of the eigenvalues of the Laplacian on a domain $Ω$ of finite measure are bounded in terms of their semiclassical limit expressions. We show that these inequalities can be improved by a multiplicative factor that depends only on the dimension and the product $\sqrtΛ|Ω|^{1/d}$, where $Λ$ is the eigenvalue cut-off parameter in the definition of the Riesz mean. The same holds when $|Ω|^{1/d}$ is replaced by a generalized inradius of $Ω$. Finally, we show similar inequalities in two dimensions in the presence of a constant magnetic field. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_02388 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Improved semiclassical eigenvalue estimates for the Laplacian and the Landau Hamiltonian Frank, Rupert L. Larson, Simon Pfeiffer, Paul Spectral Theory Mathematical Physics The Berezin--Li--Yau and the Kröger inequalities show that Riesz means of order $\geq 1$ of the eigenvalues of the Laplacian on a domain $Ω$ of finite measure are bounded in terms of their semiclassical limit expressions. We show that these inequalities can be improved by a multiplicative factor that depends only on the dimension and the product $\sqrtΛ|Ω|^{1/d}$, where $Λ$ is the eigenvalue cut-off parameter in the definition of the Riesz mean. The same holds when $|Ω|^{1/d}$ is replaced by a generalized inradius of $Ω$. Finally, we show similar inequalities in two dimensions in the presence of a constant magnetic field. |
| title | Improved semiclassical eigenvalue estimates for the Laplacian and the Landau Hamiltonian |
| topic | Spectral Theory Mathematical Physics |
| url | https://arxiv.org/abs/2502.02388 |