On Sharp Heisenberg Uncertainty Principle and the stability

TL;DR

This work develops a linearized, function-space framework for sharp Heisenberg-type uncertainty principles by combining spherical harmonic decomposition with Hardy inequalities. It proves a sharp Hydrogen Uncertainty Principle in dimensions $N\ge 4$ with explicit extremals, and provides refined bounds for $N=2,3$, including existence of minimizers for the best constants. The authors also derive exact stability constants and extremals for two related HUP inequalities, establishing precise quantitative stability in terms of distances to the extremal cone and offering explicit extremal constructions via hypergeometric functions. The results have direct implications for the sharpness and stability of uncertainty-type inequalities in PDEs and harmonic analysis, with clear formulas and extremal profiles for high- and low-dimensional cases.

Abstract

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 $\|Δu\|_2 \|r\nabla u\|_2 - \frac{N+2}{2}\|\nabla u\|^2_2$ in $\mathbb{R}^N$ ($N \geq 2$), studied recently by Duong-Nguyen [CVPDE, 2025] and Do-Lam-Lu-Zhang [arXiv:2505.02758v1].

Theorems & Definitions (13)

• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 1.5
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 3.1
• Proposition 3.2