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Semi-classical limit of an attractive Fermi gas in one or two dimensions

Thomas Gamet

Abstract

We study the ground-state of a Fermi gas with short range attrative interactions in one or two dimensions. N fermions are placed in a confining potential, and interact with each other through a negative potential, whose range is larger than the typical distance between particles. We show the convergence of the ground state energy of the Hamiltonian to a Thomas-Fermi energy in the large N limit. Furthermore, we prove convergence of the ground states, in the sense of their Husimi functions.

Semi-classical limit of an attractive Fermi gas in one or two dimensions

Abstract

We study the ground-state of a Fermi gas with short range attrative interactions in one or two dimensions. N fermions are placed in a confining potential, and interact with each other through a negative potential, whose range is larger than the typical distance between particles. We show the convergence of the ground state energy of the Hamiltonian to a Thomas-Fermi energy in the large N limit. Furthermore, we prove convergence of the ground states, in the sense of their Husimi functions.
Paper Structure (19 sections, 32 theorems, 338 equations)

This paper contains 19 sections, 32 theorems, 338 equations.

Table of Contents

  1. Introduction
  2. Context
  3. Convergence of the energy
  4. Convergence of Husimi functions
  5. Short plan of the paper
  6. Upper bound on the energy
  7. Reduction to the Hartree energy
  8. Estimate of the Hartree energy
  9. Lower bound on the energy
  10. A priori bounds
  11. Semi-classical computations
  12. The Diaconis-Freedman theorem and its direct consequences
  13. Averaging the measure
  14. Conclusion
  15. Convergence of states
  16. ...and 4 more sections

Key Result

Theorem 1.8

Let $d=1$ or $2$, and $0 < \beta < \frac{2}{d(2d + 1)}$. If $V$ and $w$ satisfy Assumptions assumption_V and assumption_w, we have with the notation introduced in Definitions definition_hamiltonien and definition_energies_vlasov_thomas_fermi.

Theorems & Definitions (96)

  • Definition 1.1: The Hamiltonian and its domain
  • Definition 1.2: Vlasov and Thomas-Fermi energies
  • Remark 1.3: Constraints on the dimension
  • Remark 1.4: Link between Vlasov and Thomas-Fermi energies
  • Theorem 1.8: Ground-state energy
  • Remark 1.9: Constraint on $\beta$
  • Remark 1.10: Assumptions on $w$ and $V$
  • Definition 1.11: Space and momentum density
  • Definition 1.12: Reduced density matrices
  • Definition 1.13: Coherent state
  • ...and 86 more