Improved leading coefficient in the individual Berezin-Li-Yau bound via energy orthogonality
Abstract
Let λk be the kth eigenvalue of the Dirichlet Laplacian on an open set Ω⊂ℝn of finite positive measure. The direct individual consequence of the Berezin--Li--Yau sum inequality has leading coefficient n/(n+2) relative to the Weyl term. The main contribution of this paper is a strict improvement of this coefficient. Energy orthogonality gives a frequency-dependent cap on the Fourier density of the first k eigenfunctions. Combining this cap with the standard Bessel bound and a bathtub principle for the radial capacity yields λk≥ cn(2π)2ωn−2/n |Ω|−2/nk2/n, n/(n+2)<cn<1, for every k≥1 and every n≥2, without boundary regularity. The constants are characterized by explicit scalar equations; in dimension two, c2=0.5383068077…, giving a 7.66% improvement over the individual Li--Yau coefficient. We emphasize that this improves the leading coefficient in the individual eigenvalue bound and the new constant cn is independent of the geometry and index k.
Related
- Improved Berezin--Li--Yau Bounds for Dirichlet Eigenvalues via Fourier Coupling and Frequency Concentration
- Toward Pólya's Conjecture: Improving the Individual Li-Yau Bound via Energy Orthogonality
- Convexity of the Berezin range of composition operators induced by automorphisms of the Unit Ball in ℂn
- Berezin Transform Commutators
- Products of Berezin Transforms on Generalized Fock Spaces
BibTeX
@misc{wang2026improved,
title = {{Improved leading coefficient in the individual Berezin-Li-Yau bound via energy orthogonality}},
author = {Yifan Wang and Hehu Xie},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23923},
eprint = {2609.23923},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23923},
note = {CC0},
}
RIS
TY - UNPB
TI - Improved leading coefficient in the individual Berezin-Li-Yau bound via energy orthogonality
AU - Yifan Wang
AU - Hehu Xie
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23923
UR - https://arxiv.org/abs/2609.23923
AB - Let λ_k be the kth eigenvalue of the Dirichlet Laplacian on an open set Ω⊂ℝ^n of finite positive measure. The direct individual consequence of the Berezin--Li--Yau sum inequality has leading coefficient n/(n+2) relative to the Weyl term. The main contribution of this paper is a strict improvement of this coefficient. Energy orthogonality gives a frequency-dependent cap on the Fourier density of the first k eigenfunctions. Combining this cap with the standard Bessel bound and a bathtub principle for the radial capacity yields λ_k≥ c_n(2π)^2ω_n^−2/n |Ω|^−2/nk^2/n, n/(n+2)<c_n<1, for every k≥1 and every n≥2, without boundary regularity. The constants are characterized by explicit scalar equations; in dimension two, c_2=0.5383068077…, giving a 7.66% improvement over the individual Li--Yau coefficient. We emphasize that this improves the leading coefficient in the individual eigenvalue bound and the new constant c_n is independent of the geometry and index k.
ER -