Sharp stability for the second-order weighted Heisenberg Uncertainty Principle with explicit stability constants and optimizers

Abstract

By using spherical harmonicas decomposition and Gaussian-type Poincaré inequalities, we establish several sharp stability estimates for the following second-order weighted Heisenberg Uncertainty Principle \begin{equation*} \int_{\mathbb{R}^N} \!\frac{|Δu|^2} {|x|^{2α}} \mathrm{d}x \int_{\mathbb{R}^N} |x|^{2α+2}|\nabla u|^2\mathrm{d}x \geq \frac{(N+4α+2)^2}{4} \left(\int_{\mathbb{R}^N} |\nabla u|^2 \mathrm{d}x\right)^2, \end{equation*} and \begin{equation*} \int_{\mathbb{R}^{N}} \frac{|Δu|^{2}} {|x|^{2α}} \mathrm{d}x +\int_{\mathbb{R}^{N}} \left|x\right|^{2α+2} |\nabla u|^{2}\mathrm{d}x \ge\left(N+4α+2\right) \int_{\mathbb{R}^{N}} |\nabla u|^{2}\mathrm{d}x. \end{equation*} We also provide the explicit value and the necessary and sufficient condition for attainability of the sharp stability constants. Moreover, when $α=0$, our results reduce into those of [\emph{Calc. Var. Partial Differential Equations} \textbf{64} (2025), Paper No. 129], [\emph{J. Funct. Anal.} \textbf{290} (2026), Paper No. 111321] and [arXiv:2510.00453].