Improved Berezin--Li--Yau Bounds for Dirichlet Eigenvalues via Fourier Coupling and Frequency Concentration
Abstract
Let 0<λ1≤λ2≤⋯ be the Dirichlet eigenvalues of an open set Ω⊂ℝn, n≥2. We develop two Fourier-space refinements of the Berezin--Li--Yau argument. For every finite-measure open set, coordinate commutator identities and coupled scalar--vector projections lead to a one-dimensional variational constant Cnnew and the individual bound λk≥ Cnnew(2π)2ωn−2/n |Ω|−2/nk2/n, n/(n+2)<cnenv<Cnnew<1. Here n/(n+2) is the classical Li--Yau individual coefficient and cnenv is the coefficient associated with the uncoupled scalar envelope developed in our previous paper. Thus the displayed strict hierarchy separates the gain of the scalar energy envelope from the further gain supplied by the commutator coupling. For bounded open sets, a frequency-concentration operator, its first two traces, and the Ky Fan principle force Fourier mass outside the Li--Yau filling ball. A two-region bathtub argument then gives ∑j=1kλj ≥ 𝒞k(2)(Ω) n/(n+2)(2π)2ωn−2/n |Ω|−2/nk1+2/n, 𝒞k(2)(Ω)>1, where 𝒞k(2)(Ω) is strictly decreasing in k and converges to one, and 𝒞k(2)(Ω)−1≥ (n+2)/(2(n+1)2) ((ωn−1)/ωn)2 (3/4)n−1k−2/n. Thus the sharp leading term in the Berezin--Li--Yau sum is supplemented by an explicit positive, volume-only correction of order k. Apart from boundedness in the concentration argument, no boundary regularity, convexity, or connectedness is assumed. These improvements do not prove Pólya's conjecture: the individual coefficient is below one, and the sum multiplier tends to one.
Related
- Arithmetic classification of the Dirichlet eigenvalues for a one-dimensional nonlocal p-Laplacian problem
- Inequalities between Dirichlet and Neumann eigenvalues in large dimensions
- Sums of Dirichlet eigenvalues in proximity of the disk and the ball
- Faber-Krahn inequality for the first Dirichlet eigenvalues of the logarithmic Hardy operator
- Spectral analysis and shape optimization of Steklov--Dirichlet eigenvalues on shell-type domains
BibTeX
@misc{wang2026improved,
title = {{Improved Berezin--Li--Yau Bounds for Dirichlet Eigenvalues via Fourier Coupling and Frequency Concentration}},
author = {Yifan Wang and Hehu Xie},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01741},
eprint = {2611.01741},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01741},
note = {CC0},
}
RIS
TY - UNPB
TI - Improved Berezin--Li--Yau Bounds for Dirichlet Eigenvalues via Fourier Coupling and Frequency Concentration
AU - Yifan Wang
AU - Hehu Xie
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01741
UR - https://arxiv.org/abs/2611.01741
AB - Let 0<λ_1≤λ_2≤⋯ be the Dirichlet eigenvalues of an open set Ω⊂ℝ^n, n≥2. We develop two Fourier-space refinements of the Berezin--Li--Yau argument. For every finite-measure open set, coordinate commutator identities and coupled scalar--vector projections lead to a one-dimensional variational constant C_n^new and the individual bound λ_k≥ C_n^new(2π)^2ω_n^−2/n |Ω|^−2/nk^2/n, n/(n+2)<c_n^env<C_n^new<1. Here n/(n+2) is the classical Li--Yau individual coefficient and c_n^env is the coefficient associated with the uncoupled scalar envelope developed in our previous paper. Thus the displayed strict hierarchy separates the gain of the scalar energy envelope from the further gain supplied by the commutator coupling. For bounded open sets, a frequency-concentration operator, its first two traces, and the Ky Fan principle force Fourier mass outside the Li--Yau filling ball. A two-region bathtub argument then gives ∑_j=1^kλ_j ≥ 𝒞_k^(2)(Ω) n/(n+2)(2π)^2ω_n^−2/n |Ω|^−2/nk^1+2/n, 𝒞_k^(2)(Ω)>1, where 𝒞_k^(2)(Ω) is strictly decreasing in k and converges to one, and 𝒞_k^(2)(Ω)−1≥ (n+2)/(2(n+1)^2) ((ω_n−1)/ω_n)^2 (3/4)^n−1k^−2/n. Thus the sharp leading term in the Berezin--Li--Yau sum is supplemented by an explicit positive, volume-only correction of order k. Apart from boundedness in the concentration argument, no boundary regularity, convexity, or connectedness is assumed. These improvements do not prove Pólya's conjecture: the individual coefficient is below one, and the sum multiplier tends to one.
ER -