标题： 关于精确的海森堡不确定性原理和稳定性 显示英文标题

标题： On Sharp Heisenberg Uncertainty Principle and the stability

摘要： 在本工作中，我们总结了线性化方法以研究海森堡不确定性原理，并解释了相同的方法可以用于处理稳定性问题。作为应用示例，结合球面调和分解和Hardy不等式，我们修订了两个不等式族。我们首先在四维情况下对Cazacu-Flynn-Lam的猜想[JFA, 2022]给出了肯定回答，该猜想涉及精确的氢原子不确定性原理，并改进了Chen-Tang [arXiv:2508.15221v1]在$\mathbb{R}^2$和$\mathbb{R}^3$中的最近估计。另一方面，我们确定了与$\|\Delta u\|_2 \|r\nabla u\|_2 - \frac{N+2}{2}\|\nabla u\|^2_2$在$\mathbb{R}^N$($N \geq 2$) 相关的两个稳定性估计的最佳常数和极值函数，这些估计最近由Duong-Nguyen [CVPDE, 2025] 和 Do-Lam-Lu-Zhang [arXiv:2505.02758v1] 研究过。 显示英文摘要

摘要： In this work, we summarize the linearization method to study the Heisenberg Uncertainty Principles, and explain that the same approach can be used to handle the stability problem. As examples of application, combining with spherical harmonic decomposition and the Hardy inequalities, we revise two families of inequalities. We give firstly an affirmative answer in dimension four to Cazacu-Flynn-Lam's conjecture [JFA, 2022] for the sharp Hydrogen Uncertainty Principle, and improve the recent estimates of Chen-Tang [arXiv:2508.15221v1] in $\mathbb{R}^2$ and $\mathbb{R}^3$. On the other hand, we identify the best constants and extremal functions for two stability estimates associated to $\|\Delta u\|_2 \|r\nabla u\|_2 - \frac{N+2}{2}\|\nabla u\|^2_2$ in $\mathbb{R}^N$ ($N \geq 2$), studied recently by Duong-Nguyen [CVPDE, 2025] and Do-Lam-Lu-Zhang [arXiv:2505.02758v1].