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Semiclassical equivalence of two white dwarf models as ground states of the relativistic Hartree-Fock and Vlasov-Poisson energies

Younghun Hong, Sangdon Jin, Jinmyoung Seok

Abstract

We are concerned with the semi-classical limit for ground states of the relativistic Hartree-Fock energies (HF) under a mass constraint, which are considered as the quantum mean-field model of white dwarfs \cite{LeLe}. In Jang and Seok \cite{JS}, fermionic ground states of the relativistic Vlasov-Poisson energy (VP) are constructed as a classical mean-field model of white dwarfs, and are shown to be equivalent to the classical Chandrasekhar model. In this paper, we prove that as the reduced Planck constant $\hbar$ goes to the zero, the $\hbar$-parameter family of the ground energies and states of (HF) converges to the fermionic ground energy and state of (VP) with the same mass constraint.

Semiclassical equivalence of two white dwarf models as ground states of the relativistic Hartree-Fock and Vlasov-Poisson energies

Abstract

We are concerned with the semi-classical limit for ground states of the relativistic Hartree-Fock energies (HF) under a mass constraint, which are considered as the quantum mean-field model of white dwarfs \cite{LeLe}. In Jang and Seok \cite{JS}, fermionic ground states of the relativistic Vlasov-Poisson energy (VP) are constructed as a classical mean-field model of white dwarfs, and are shown to be equivalent to the classical Chandrasekhar model. In this paper, we prove that as the reduced Planck constant goes to the zero, the -parameter family of the ground energies and states of (HF) converges to the fermionic ground energy and state of (VP) with the same mass constraint.
Paper Structure (19 sections, 17 theorems, 118 equations)

This paper contains 19 sections, 17 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Backgrounds and setup
  3. Relativistic Hartree-Fock formulation
  4. Relativistic Vlasov-Poisson formulation
  5. Statement of the main result
  6. Notations
  7. Outline of the paper
  8. Acknowledgement
  9. Key inequalities and Critical masses
  10. Potential energy estimates
  11. Exchange term estimates
  12. Critical masses
  13. Quantum and kinetic white dwarfs and their basic properties
  14. Relativistic Weyl's law
  15. Setup
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $\hbar>0$. For $0<M<\mathbf{M}_{\textup{qm}}$, the following hold.

Theorems & Definitions (34)

  • Theorem 1.1: $LeLe$
  • Remark 1.2
  • Theorem 1.3: $JS$
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Lemma 2.1: Kinetic interpolation inequality; endpoint case
  • proof
  • Lemma 2.2: Lieb-Thirring inequality
  • proof
  • ...and 24 more