The Sharp Constant in the Second-Order Hydrogen Uncertainty Principle: Spectral Characterization and Quantitative Approximation
Abstract
In this work, we study the sharp constant ℋd in the second-order Hydrogen Uncertainty Principle in every dimension d≥2. Using a spectral basis built from generalized Laguerre polynomials, together with a linearization procedure and spherical harmonic decomposition, we establish a spectral characterization of ℋd. More precisely, we show that ℋd = 1/2min{d+1,cd*}, where cd*∈(0,d+3) is the unique parameter such that |Scd*(d)|ℓ2→ℓ2=1. Here, for 0<c<d+3, Sc(d) is an explicit positive Hilbert--Schmidt operator on ℓ2(ℕ0). 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 cd*. For d≥4, we recover the known identity ℋd=(d+1)/2 through a spectral argument, giving an alternative proof of the results of Cazacu, Flynn and Lam \cite{CazacuFlynnLam} for d≥5 and Huang and Ye \cite{HuangYe} for d=4. For d∈{2,3}, this characterization yields ℋd=cd*/2. We then exploit this spectral formulation to derive an explicit finite recursive scheme yielding effective two-sided bounds for ℋ2 and ℋ3, substantially improving the estimates of Huang and Ye \cite{HuangYe}.
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BibTeX
@misc{jleli2026sharp,
title = {{The Sharp Constant in the Second-Order Hydrogen Uncertainty Principle: Spectral Characterization and Quantitative Approximation}},
author = {Mohamed Jleli and Bessem Samet},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34916},
eprint = {2610.34916},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34916},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Sharp Constant in the Second-Order Hydrogen Uncertainty Principle: Spectral Characterization and Quantitative Approximation
AU - Mohamed Jleli
AU - Bessem Samet
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.34916
UR - https://arxiv.org/abs/2610.34916
AB - In this work, we study the sharp constant ℋ_d in the second-order Hydrogen Uncertainty Principle in every dimension d≥2. Using a spectral basis built from generalized Laguerre polynomials, together with a linearization procedure and spherical harmonic decomposition, we establish a spectral characterization of ℋ_d. More precisely, we show that ℋ_d = 1/2min{d+1,c_d^*}, where c_d^*∈(0,d+3) is the unique parameter such that |S_c_d^*^(d)|_ℓ^2→ℓ^2=1. Here, for 0<c<d+3, S_c^(d) is an explicit positive Hilbert--Schmidt operator on ℓ^2(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≥4, we recover the known identity ℋ_d=(d+1)/2 through a spectral argument, giving an alternative proof of the results of Cazacu, Flynn and Lam \cite{CazacuFlynnLam} for d≥5 and Huang and Ye \cite{HuangYe} for d=4. For d∈{2,3}, this characterization yields ℋ_d=c_d^*/2. We then exploit this spectral formulation to derive an explicit finite recursive scheme yielding effective two-sided bounds for ℋ_2 and ℋ_3, substantially improving the estimates of Huang and Ye \cite{HuangYe}.
ER -