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arXiv · 2403.12640

Hardy inequalities for large fermionic systems

Rupert L. Frank·Thomas Hoffmann-Ostenhof·Ari Laptev·Jan Philip Solovej
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Abstract

Given $0<s<\frac d2$ with $s\leq 1$, we are interested in the large $N$-behavior of the optimal constant $\kappa_N$ in the Hardy inequality $\sum_{n=1}^N (-\Delta_n)^s \geq \kappa_N \sum_{n<m} |X_n-X_m|^{-2s}$, when restricted to antisymmetric functions. We show that $N^{1-\frac{2s}d}\kappa_N$ has a positive, finite limit given by a certain variational problem, thereby generalizing a result of Lieb and Yau related to the Chandrasekhar theory of gravitational collapse.

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Rupert L. Frank, Thomas Hoffmann-Ostenhof, Ari Laptev, Jan Philip Solovej. 2024-03-19. Hardy inequalities for large fermionic systems. https://arxiv.org/abs/2403.12640

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