On Sharp Heisenberg Uncertainty Principle and the stability

2025/10/01 by Xia Huang, Huang, Xia, Dong Ye +1
Decision Sciences · #Analysis of PDEs (math.AP) #FOS: Mathematics #Probabilistic and Robust Engineering Design

paper · pdf · doi:10.48550/arxiv.2510.00453

openalex publication_date 2025/10/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ℝ² and ℝ³. On the other hand, we identify the best constants and extremal functions for two stability estimates associated to ‖Δu‖₂ ‖r∇ u‖₂ - (N+2)/(2)‖∇ u‖²₂ in ℝ^N (N ≥ 2), studied recently by Duong-Nguyen [CVPDE, 2025] and Do-Lam-Lu-Zhang [arXiv:2505.02758v1].

Citations

• On the extremal functions of second order uncertainty principles: symmetry and symmetry breaking
• Sharp stability of the Heisenberg Uncertainty Principle: Second-Order and Curl-Free Field Cases
• On a minimisation problem related to the solenoidal uncertainty
• Caffarelli-Kohn-Nirenberg identities, inequalities and their stabilities
• First order Hardy inequalities revisited
• Sharp second order uncertainty principles

Related

• Sharp stability of the Heisenberg Uncertainty Principle: Second-Order and Curl-Free Field Cases
• On the extremal functions of second order uncertainty principles: symmetry and symmetry breaking
• Sharp second order uncertainty principles
• The Generalized Uncertainty Principle. New Bounds and Trends
• Sharp uncertainty principles on Riemannian manifolds: the influence of curvature