Toward Pólya's Conjecture: Improving the Individual Li-Yau Bound via Energy Orthogonality
Abstract
We establish two complementary lower-bound mechanisms for individual eigenvalues of the Dirichlet Laplacian. First, energy orthogonality yields a frequency-dependent cap on the Fourier density of a finite spectral projection. Combining this cap with the standard L2 Bessel estimate and a radial-capacity bathtub principle gives, on every open set of finite positive measure in ℝn with n≥2, λk≥ cn(2π)2ωn−2/n|Ω|−2/nk2/n, n/(n+2)<cn<1. The constant cn is characterized by a scalar equation, with c2=0.5383068077…. This estimate preserves Weyl scaling and strictly improves the individual consequence of the Li-Yau sum inequality, although it does not improve the sharp leading coefficient in that sum inequality. Second, we retain part of the spectral deficit discarded when an eigenvalue sum is bounded by its largest term. A lower bound for the counting function, integrated through the exact first Riesz-mean identity, leads to a strictly monotone scalar equation. Its unique positive root is no weaker than the volume-only bound, and we give necessary and sufficient criteria for strict improvement over both that baseline and any independent lower bound. Quantitative estimates of Jiang-Lin provide an explicit implementation on bounded Lipschitz domains. The final comparisons and numerical example distinguish improvements within this framework from stronger estimates available under additional geometric or spectral assumptions.
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BibTeX
@misc{wang2026toward,
title = {{Toward Pólya's Conjecture: Improving the Individual Li-Yau Bound via Energy Orthogonality}},
author = {Yifan Wang and Hehu Xie},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17219},
eprint = {2609.17219},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17219},
note = {CC0},
}
RIS
TY - UNPB
TI - Toward Pólya's Conjecture: Improving the Individual Li-Yau Bound via Energy Orthogonality
AU - Yifan Wang
AU - Hehu Xie
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17219
UR - https://arxiv.org/abs/2609.17219
AB - We establish two complementary lower-bound mechanisms for individual eigenvalues of the Dirichlet Laplacian. First, energy orthogonality yields a frequency-dependent cap on the Fourier density of a finite spectral projection. Combining this cap with the standard L^2 Bessel estimate and a radial-capacity bathtub principle gives, on every open set of finite positive measure in ℝ^n with n≥2, λ_k≥ c_n(2π)^2ω_n^−2/n|Ω|^−2/nk^2/n, n/(n+2)<c_n<1. The constant c_n is characterized by a scalar equation, with c_2=0.5383068077…. This estimate preserves Weyl scaling and strictly improves the individual consequence of the Li-Yau sum inequality, although it does not improve the sharp leading coefficient in that sum inequality. Second, we retain part of the spectral deficit discarded when an eigenvalue sum is bounded by its largest term. A lower bound for the counting function, integrated through the exact first Riesz-mean identity, leads to a strictly monotone scalar equation. Its unique positive root is no weaker than the volume-only bound, and we give necessary and sufficient criteria for strict improvement over both that baseline and any independent lower bound. Quantitative estimates of Jiang-Lin provide an explicit implementation on bounded Lipschitz domains. The final comparisons and numerical example distinguish improvements within this framework from stronger estimates available under additional geometric or spectral assumptions.
ER -