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arXiv · 2508.15221

On the extremal functions of second order uncertainty principles: symmetry and symmetry breaking

Xiao-Ping Chen·Chun-Lei Tang
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Abstract

This paper focus on the symmetry and symmetry breaking about the second order Hydrogen Uncertainty Principle. \emph{Firstly}, by choosing a suitable test function, we give a negative answer to the conjecture presented by Cazacu, Flynn and Lam in [\emph{J. Funct. Anal.} \textbf{283} (2022), Paper No. 109659, 37 pp] for $N\in\{2,3\}$, and emphasizing the symmetry breaking phenomenon. \emph{Secondly}, we obtain a family of sharp weighted second order Hydrogen Uncertainty Principle, and prove the extremal functions are radial, which extends the work of Duong and Nguyen [The sharp second order Caffareli-Kohn-Nirenberg inequality and stability estimates for the sharp second order uncertainty principle, arXiv:2102.01425].

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Xiao-Ping Chen, Chun-Lei Tang. 2025-08-21. On the extremal functions of second order uncertainty principles: symmetry and symmetry breaking. https://arxiv.org/abs/2508.15221

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