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Hardy inequalities for large fermionic systems

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

Abstract

Given $0<s<\frac d2$ with $s\leq 1$, we are interested in the large $N$-behavior of the optimal constant $κ_N$ in the Hardy inequality $\sum_{n=1}^N (-Δ_n)^s \geq κ_N \sum_{n<m} |X_n-X_m|^{-2s}$, when restricted to antisymmetric functions. We show that $N^{1-\frac{2s}d}κ_N$ 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.

Hardy inequalities for large fermionic systems

Abstract

Given with , we are interested in the large -behavior of the optimal constant in the Hardy inequality , when restricted to antisymmetric functions. We show that 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.
Paper Structure (17 sections, 13 theorems, 147 equations)

This paper contains 17 sections, 13 theorems, 147 equations.

Table of Contents

  1. Introduction and main result
  2. Lower bound
  3. An electrostatic inequality
  4. Lieb--Thirring inequality
  5. Coherent states
  6. Summary so far
  7. Domination of the nearest neighbor attraction
  8. Proof of the lower bound in Theorem \ref{['main']}
  9. The case without antisymmetry
  10. Upper bound
  11. A bound on the indirect part of the Riesz energy
  12. Relaxation to density matrices
  13. Construction of $\gamma$ using coherent states
  14. The semiclassical problem
  15. Proof of the upper bound in Theorem \ref{['main']}
  16. ...and 2 more sections

Key Result

Theorem 1

Let $d\geq 1$ and $0<s<\frac{d}{2}$ with $s\leq 1$. Then

Theorems & Definitions (24)

  • Theorem 1
  • Proposition 3
  • proof
  • Lemma 4
  • Lemma 5
  • proof
  • Corollary 6
  • proof
  • Proposition 7
  • Remark 8
  • ...and 14 more