Sharp stability of the Heisenberg Uncertainty Principle: Second-Order and Curl-Free Field Cases
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866917158711197696 |
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| author | Do, Anh Xuan Lam, Nguyen Lu, Guozhen |
| author_facet | Do, Anh Xuan Lam, Nguyen Lu, Guozhen |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_02758 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sharp stability of the Heisenberg Uncertainty Principle: Second-Order and Curl-Free Field Cases Do, Anh Xuan Lam, Nguyen Lu, Guozhen Analysis of PDEs Mathematical Physics Classical Analysis and ODEs 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. |
| title | Sharp stability of the Heisenberg Uncertainty Principle: Second-Order and Curl-Free Field Cases |
| topic | Analysis of PDEs Mathematical Physics Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2505.02758 |