AI 中文总结
研究为前\(d\)个非零诺伊曼特征值的倒数和等周不等式建立定量稳定性，证明倒数和亏缺能控制特征值相关指标，近球形扰动显示指数最优，二维中矩阵方法改进了对前两个非零特征值联合谱图像的约束。
AI 中文摘要
我们为He、Li和Tang最近证明的前\(d\)个非零诺伊曼特征值的倒数和等周不等式建立了定量稳定性。证明了对于\(\mathbb{R}^d\)中的有界利普希茨区域，倒数和亏缺二次控制归一化特征值位移和弗伦克尔不对称性，同时线性控制特征值和的位移。进一步结果表明，经典的塞格–温伯格亏缺控制前两个非零特征值之间的差距和第一个特征值簇的全宽。近球形扰动表明所有稳定性指数都是最优的。在二维中，相同的矩阵方法对前两个非零特征值的联合谱图像产生了改进的约束。
英文摘要
We establish quantitative stability for the reciprocal-sum isoperimetric inequality for the first $d$ nonzero Neumann eigenvalues, recently proved by He, Li, and Tang. We prove that for bounded Lipschitz domains in $\mathbb{R}^d$, the reciprocal-sum deficit controls quadratically both the normalized eigenvalue displacements and the Fraenkel asymmetry, while controlling the displacement of the eigenvalue sum linearly. A further result shows that the classical Szegő--Weinberger deficit controls both the gap between the first two nonzero eigenvalues and the full width of the first eigenvalue cluster. Nearly spherical perturbations demonstrate that all the stability exponents are optimal. In dimension two, the same matrix method yields improved constraints on the joint spectral image of the first two nonzero eigenvalues.
