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arXiv · 2502.02388

Improved semiclassical eigenvalue estimates for the Laplacian and the Landau Hamiltonian

Rupert L. Frank·Simon Larson·Paul Pfeiffer
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Abstract

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.

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Rupert L. Frank, Simon Larson, Paul Pfeiffer. 2025-12-06. Improved semiclassical eigenvalue estimates for the Laplacian and the Landau Hamiltonian. https://arxiv.org/abs/2502.02388

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