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Commutator Estimates for Low-Temperature Fermi Gases

Jacky J. Chong, Laurent Lafleche, Jinyeop Lee, Chiara Saffirio

Abstract

We investigate the semiclassical regularity of thermal equilibria in the presence of a harmonic potential at low temperature; that is, we obtain the asymptotic behavior of the Schatten norms of commutators of the one-body operators associated with these equilibria and the position and momentum operators. We also obtain upper bounds in the magnetic field case for the Fock-Darwin Hamiltonian. Our estimates, in particular, allow us to observe several regimes depending on the joint behavior of the Planck constant, the temperature, and the strength of the magnetic field.

Commutator Estimates for Low-Temperature Fermi Gases

Abstract

We investigate the semiclassical regularity of thermal equilibria in the presence of a harmonic potential at low temperature; that is, we obtain the asymptotic behavior of the Schatten norms of commutators of the one-body operators associated with these equilibria and the position and momentum operators. We also obtain upper bounds in the magnetic field case for the Fock-Darwin Hamiltonian. Our estimates, in particular, allow us to observe several regimes depending on the joint behavior of the Planck constant, the temperature, and the strength of the magnetic field.

Paper Structure

This paper contains 14 sections, 17 theorems, 200 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Main results
  4. Motivation and known results
  5. Structure of the paper
  6. Size of Commutators: Harmonic Oscillator
  7. Classical case
  8. Quantum case
  9. Size of Commutators: with Magnetic Field
  10. Classical case
  11. Quantum case
  12. Upper Bounds for the Size of Commutators
  13. Equilibrium at zero temperature
  14. Equilibrium at positive temperature

Key Result

Theorem 1.1

Let $F(r) = (1+e^{\beta(r-\mu)})^{-1}$ and the corresponding one-particle density operator ${\boldsymbol{\gamma}}_{\beta, \mu} := F(\left\lvert \mathbf{z} \right\rvert^2)$, with $\mathbf{z} = (x,\boldsymbol{p})$. Fix $\mu\in \mathbb R$ and $p\in [1, \infty)$. If $\hbar \in (0, 1]$ and $\beta\hbar \l If $\hbar \in (0, 1]$ and $\beta\hbar\ge 1$, then $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (42)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Theorem 1.8
  • Remark 1.9
  • Lemma 2.1
  • ...and 32 more