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Norm convergence of confined fermionic systems at zero temperature

Esteban Cárdenas

Abstract

The semi-classical limit of ground states of large systems of fermions was studied by Fournais, Lewin and Solovej in (Calc. Var. Partial Differ. Equ., 2018). In particular, the authors prove weak convergence towards classical states associated to the minimizers of the Thomas-Fermi functional. In this paper, we revisit this limit, and show that under certain assumptions--and, using simple arguments--it is possible to prove that strong convergence holds true in relevant normed spaces.

Norm convergence of confined fermionic systems at zero temperature

Abstract

The semi-classical limit of ground states of large systems of fermions was studied by Fournais, Lewin and Solovej in (Calc. Var. Partial Differ. Equ., 2018). In particular, the authors prove weak convergence towards classical states associated to the minimizers of the Thomas-Fermi functional. In this paper, we revisit this limit, and show that under certain assumptions--and, using simple arguments--it is possible to prove that strong convergence holds true in relevant normed spaces.
Paper Structure (15 sections, 9 theorems, 90 equations)

This paper contains 15 sections, 9 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Thomas-Fermi theory
  3. Convergence of states
  4. Our contribution
  5. Main results
  6. Assumptions, definitions and notations
  7. Main results
  8. Proof of the main results
  9. Proof of Theorem \ref{['theorem 1']}
  10. Proof of Theorem \ref{['thm 2']}
  11. Application: the harmonic trap
  12. Results on $k$-particle distribution functions
  13. Definitions
  14. Statements
  15. Proof of Theorem \ref{['thm 3']}

Key Result

Theorem 1

Let $f_N$, $m_N$ and $f$ be as in states, and let Assumption assumption 1 hold. Then, the following statements are true.

Theorems & Definitions (33)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 1
  • Remark 2.4
  • Theorem 1: Norm convergence
  • Remark 2.5: $k$-particle functions
  • Remark 2.6: Positive temperature
  • Remark 2.7: Position densities
  • Corollary 2.1
  • ...and 23 more