The Sharp Constant in the Second-Order Hydrogen Uncertainty Principle: Spectral Characterization and Quantitative Approximation

In this work, we study the sharp constant \(\mathcal H_d\) in the second-order Hydrogen Uncertainty Principle in every dimension \(d\ge2\). Using a spectral basis built from generalized Laguerre polynomials, together with a linearization procedure and spherical harmonic decomposition, we establish a spectral characterization of \(\mathcal H_d\). More precisely, we show that \[ \mathcal H_d = \frac12\min\{d+1,c_d^*\}, \] where \(c_d^*\in(0,d+3)\) is the unique parameter such that \[ \|S_{c_d^*}^{(d)}\|_{\ell^2\to\ell^2}=1. \] Here, for \(0