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

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 $\alpha =0$, our results reduce into those of [\emph{Calc. Var. Partial Differential Equations} $\mathbf{\text{64}}$ (2025), Paper No. 129], [\emph{J. Funct. Anal.} $\mathbf{\text{290}}$ (2026), Paper No. 111321] and [arXiv:2510.00453].