Hardy inequalities for large fermionic systems

Creators: Frank, Rupert L.¹; Hoffmann-Ostenhof, Thomas²; Laptev, Ari^{3, 4}; Solovej, Jan Philip⁵

1. California Institute of Technology
2. University of Vienna
3. Imperial College London
4. Sirius University of Science and Technology
5. University of Copenhagen

Abstract

Given $0 < s < 2 d$ with $s \leq 1$ , we are interested in the large $N$ -behavior of the optimal constant $κ_{N}$ in the Hardy inequality $\sum_{n = 1 N} (- Δ_{n})^{s} \geq κ_{N} \sum_{n < m} ∣ X_{n} - X_{m} ∣^{-2 s}$ , when restricted to antisymmetric functions. We show that $N^{1 - d 2 s} κ_{N}$ 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.

Copyright and License

CC-BY 4.0 European Mathematical Society. This article is published open access under our Subscribe to Open model.

Funding

RLF was partially supported by the US National Science Foundation Grant number DMS-1954995 and the DFG grants EXC-2111-390814868 and TRR 352-Project-ID 470903074. The support of the villum Centre of Excellence for the Mathematics of Quantum Theory (QMATH) Grant number 10059 to JPS is acknowledged.

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