朝向 Pólya 猜想：通过能量正交性改进个体 Li-Yau 界
Toward Pólya's Conjecture: Improving the Individual Li-Yau Bound via Energy Orthogonality
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中文总结 AI 辅助
本文通过能量正交性和谱亏量保留机制，为 Dirichlet 拉普拉斯特征值建立两个互补下界，改进个体 Li-Yau 界，并给出常数刻画与数值验证。
中文摘要 AI 辅助
我们为 Dirichlet 拉普拉斯算子的个体特征值建立了两个互补的下界机制。首先，能量正交性对有限谱投影的傅里叶密度产生一个依赖于频率的上限。将此上限与标准的 $L^2$ Bessel 估计以及径向容量浴缸原理相结合，在 $\nmathbb R^n$（$n\geq2$）中每个具有有限正测度的开集上，得到 \\[ \lambda_k\geq c_n(2\pi)^2\omega_n^{-2/n}|\Omega|^{-2/n}k^{2/n}, \qquad \frac{n}{n+2}<c_n<1. \\] 常数 $c_n$ 由标量方程刻画，其中 $c_2=0.5383068077\ldots$。该估计保持 Weyl 标度，并严格改进 Li-Yau 和不等式的个体推论，尽管它不改进该和不等式中尖锐的首项系数。其次，我们保留了当特征值之和被其最大项限制时所丢弃的部分谱亏量。通过精确的第一 Riesz 均值恒等式积分得到的计数函数下界，导致一个严格单调的标量方程。其唯一正根不弱于仅含体积的界，我们给出了严格改进该基线和任何独立下界的充分必要条件。Jiang-Lin 的定量估计在有界 Lipschitz 域上提供了显式实现。最终比较和数值示例区分了此框架内的改进与在额外几何或谱假设下可获得的更强估计。
英文摘要
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 $\mathbb R^n$ with $n\geq2$, \[ λ_k\geq c_n(2π)^2ω_n^{-2/n}|Ω|^{-2/n}k^{2/n}, \qquad \frac{n}{n+2}<c_n<1. \] The constant $c_n$ is characterized by a scalar equation, with $c_2=0.5383068077\ldots$. 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.
发表机构
- School of Statistics and Mathematics, Central University of Finance and Economics（中央财经大学统计与数学学院）
- SKLMS, NCMIS, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所）
- School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
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