Sharp stability of the Heisenberg Uncertainty Principle: Second-Order and Curl-Free Field Cases

TL;DR

The paper develops a sharp, dimension-dependent stability theory for the second-order Heisenberg Uncertainty Principle using harmonic-analysis techniques. Through a spherical-harmonics decomposition and a dimension-raising Fourier identity, it reduces second-order problems to radial analyses and derives explicit sharp constants, including $C(N)=\sqrt{N^{2}+4N-4}-N$, attained by Gaussian profiles. It also extends stability to curl-free vector fields and establishes a Gaussian-measure Poincaré-type inequality, yielding tight control over stability constants as $N$ grows large. These results improve prior bounds, provide attainability, and offer a unified, constructive framework for quantitative stability in higher-order uncertainty-type inequalities.

Abstract

Using techniques from harmonic analysis, we derive several sharp stability estimates for the second order Heisenberg Uncertainty Principle. We also present the explicit lower and upper bounds for the sharp stability constants and compute their exact limits when the dimension $N\rightarrow\infty$. Our proofs rely on spherical harmonics decomposition and Fourier analysis, differing significantly from existing approaches in the literature. Our results substantially improve the stability constants of the second order Heisenberg Uncertainty Principle recently obtained in [27]. As direct consequences of our main results, we also establish the sharp stability, with exact asymptotic behavior of the stability constants, of the Heisenberg Uncertainty Principle with curl-free vector fields and a sharp version of the second order Poincaré type inequality with Gaussian measure.

Theorems & Definitions (21)

• Theorem A
• Theorem B
• Theorem C
• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Lemma 2.1