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

Abstract

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 \(0<c<d+3\), \(S_c^{(d)}\) is an explicit positive Hilbert--Schmidt operator on \(\ell^2(\mathbb N_0)\). This characterization reveals a competition between the radial sector, which yields the value \(d+1\), and the first non-radial sector, encoded by the spectral parameter \(c_d^*\). For \(d\ge4\), we recover the known identity \[ \mathcal H_d=\frac{d+1}{2} \] through a spectral argument, giving an alternative proof of the results of Cazacu, Flynn and Lam \cite{CazacuFlynnLam} for \(d\ge5\) and Huang and Ye \cite{HuangYe} for \(d=4\). For \(d\in\{2,3\}\), this characterization yields \[ \mathcal H_d=\frac{c_d^*}{2}. \] We then exploit this spectral formulation to derive an explicit finite recursive scheme yielding effective two-sided bounds for \(\mathcal H_2\) and \(\mathcal H_3\), substantially improving the estimates of Huang and Ye \cite{HuangYe}.