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

TL;DR

This paper establishes sharp stability estimates for the second order Heisenberg Uncertainty Principle using harmonic analysis, providing explicit bounds and asymptotic behavior of stability constants, with applications to curl-free fields and Gaussian measures.

Contribution

It introduces new sharp stability estimates for the second order Heisenberg Uncertainty Principle using spherical harmonics and Fourier analysis, improving previous stability constants.

Findings

Derived explicit bounds for stability constants

Computed exact limits as dimension approaches infinity

Established sharp stability for curl-free vector fields

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 Poincar\'{e} type inequality with Gaussian measure.