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Ground state energy of the dilute spin-polarized Fermi gas: Lower bound

Asbjørn Bækgaard Lauritsen, Robert Seiringer

Abstract

We prove a lower bound on the ground state energy of the dilute spin-polarized Fermi gas capturing the leading correction to the kinetic energy resulting from repulsive interactions. This correction depends on the $p$-wave scattering length of the interaction and matches the corresponding upper bound in [J. Funct. Anal. 286.7 (2024), p. 110320].

Ground state energy of the dilute spin-polarized Fermi gas: Lower bound

Abstract

We prove a lower bound on the ground state energy of the dilute spin-polarized Fermi gas capturing the leading correction to the kinetic energy resulting from repulsive interactions. This correction depends on the -wave scattering length of the interaction and matches the corresponding upper bound in [J. Funct. Anal. 286.7 (2024), p. 110320].
Paper Structure (43 sections, 20 theorems, 217 equations)

This paper contains 43 sections, 20 theorems, 217 equations.

Table of Contents

  1. Introduction and Main Results
  2. Precise statement of results
  3. Two dimensions
  4. Second quantization
  5. Heuristics
  6. General second order perturbation theory
  7. Application to the many-body setting
  8. Rigorous Analysis
  9. Regularizing the operators
  10. The scattering function
  11. The transformation T
  12. Implementation
  13. A Priori Bounds
  14. A priori analysis of the kinetic energy
  15. A priori analysis of the interaction
  16. ...and 28 more sections

Key Result

Theorem 1.2

Let $V\in L^1$ be non-negative, radial and compactly supported. Then for $ak_F$ small enough and $N$ large enough we have

Theorems & Definitions (61)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4: Extension to less regular $V$
  • Proposition 1.5: Upper bound
  • Definition 1.6
  • Remark 1.7
  • Theorem 1.8: Two dimensions
  • Remark 1.9
  • Definition 1.13
  • ...and 51 more