发表机构
Central University of Finance and Economics; Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences; School of Mathematical Sciences, University of Chinese Academy of Sciences（中央财经大学; 中国科学院数学与系统科学研究院计算数学研究所; 中国科学院大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过能量正交性改进 Dirichlet Laplacian 个体特征值下界中的首项系数，得到优于 Li-Yau 系数的常数，且与几何和指标无关。
AI 中文摘要
设 $\lambda_k$ 为有限正测度开集 $\Omega\subset\mathbb R^n$ 上 Dirichlet Laplacian 的第 $k$ 个特征值。Berezin-Li-Yau 和不等式的直接个体推论相对于 Weyl 项具有首项系数 $n/(n+2)$。本文的主要贡献是对该系数的严格改进。能量正交性给出了前 $k$ 个特征函数的 Fourier 密度的频率相关上限。将该上限与标准 Bessel 界以及径向容量的浴缸原理相结合，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, \\] 对所有 $k\geq1$ 和所有 $n\geq2$ 成立，且无需边界正则性。这些常数由显式标量方程刻画；在二维情形，$c_2=0.5383068077\ldots$，相对于个体 Li-Yau 系数有 $7.66\\%$ 的改进。我们强调，这改进了个体特征值界中的首项系数，且新常数 $c_n$ 与几何和指标 $k$ 无关。
英文摘要
Let $λ_k$ be the $k$th eigenvalue of the Dirichlet Laplacian on an open set $Ω\subset\mathbb R^n$ 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 \[ λ_k\geq c_n(2π)^2ω_n^{-2/n} |Ω|^{-2/n}k^{2/n}, \qquad \frac{n}{n+2}<c_n<1, \] for every $k\geq1$ and every $n\geq2$, without boundary regularity. The constants are characterized by explicit scalar equations; in dimension two, $c_2=0.5383068077\ldots$, 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 $k$.
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