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Pseudo-relativistic fermionic systems with attractive Yukawa potential

Bin Chen, Yujin Guo, Phan Thành Nam, Dong Hao Ou Yang

TL;DR

The paper analyzes self-consistent fermionic systems with a pseudo-relativistic kinetic energy and an attractive Yukawa potential, proving a Chandrasekhar-type stability threshold $\lambda^{\mathrm{HFB}}(\kappa)$ that is independent of $m$ and $\theta$. It develops a rigorous variational framework for HFB and HF theories, derives exponential decay of minimizers for small Yukawa screening, and employs concentration-compactness and a Newton-type theorem to prove existence of HFB minimizers for subcritical masses and small $\theta$, while also establishing nonexistence results for HF minimizers at the critical HF coupling. Collectively, the results extend prior $\theta=0$ findings to Yukawa-corrected interactions, clarifying stability, ground-state structure, and decay properties in pseudo-relativistic fermionic systems. The findings have implications for understanding screening effects in self-gravitating quantum systems and contribute precise thresholds and qualitative behavior for ground states in nonlocal, translation-invariant variational problems.

Abstract

We study the Hartree-Fock and Hartree-Fock-Bogoliubov theories for a large fermionic system with the pseudo-relativistic kinetic energy and an attractive Yukawa interaction potential. We prove that the system is stable if and only if the total mass does not excess a critical value, and investigate the existence and properties of ground states in both sub-critical and critical mass regimes.

Pseudo-relativistic fermionic systems with attractive Yukawa potential

TL;DR

The paper analyzes self-consistent fermionic systems with a pseudo-relativistic kinetic energy and an attractive Yukawa potential, proving a Chandrasekhar-type stability threshold that is independent of and . It develops a rigorous variational framework for HFB and HF theories, derives exponential decay of minimizers for small Yukawa screening, and employs concentration-compactness and a Newton-type theorem to prove existence of HFB minimizers for subcritical masses and small , while also establishing nonexistence results for HF minimizers at the critical HF coupling. Collectively, the results extend prior findings to Yukawa-corrected interactions, clarifying stability, ground-state structure, and decay properties in pseudo-relativistic fermionic systems. The findings have implications for understanding screening effects in self-gravitating quantum systems and contribute precise thresholds and qualitative behavior for ground states in nonlocal, translation-invariant variational problems.

Abstract

We study the Hartree-Fock and Hartree-Fock-Bogoliubov theories for a large fermionic system with the pseudo-relativistic kinetic energy and an attractive Yukawa interaction potential. We prove that the system is stable if and only if the total mass does not excess a critical value, and investigate the existence and properties of ground states in both sub-critical and critical mass regimes.
Paper Structure (14 sections, 31 theorems, 334 equations)

This paper contains 14 sections, 31 theorems, 334 equations.

Table of Contents

  1. Introduction and main results
  2. Hatree--Fock--Bogoliubov theory
  3. Hartree--Fock theory
  4. Preliminary results on HFB theory
  5. Auxiliary variational problem $G_{\theta}(\lambda)$
  6. Estimates of chemical potential $\mu_{\theta}$
  7. Decaying estimate of HFB minimizers: Proof of Theorem \ref{['thm:main2']}
  8. Estimates from the mean-field equations
  9. Proof of Theorem \ref{['thm:main2']}
  10. Existence of HFB minimizers
  11. Proof of Theorem \ref{['thm:main1']}
  12. Existence and non-existence of HF minimizers
  13. Non-existence of minimizers
  14. Proof of Propositions \ref{['prop:Chandrasekhar']} and \ref{['prop:min-Gtheta']}

Key Result

Proposition 1.1

Let $m\geq 0$, $0< \kappa<4/\pi$ and $\theta\geq 0$ be given. Then there exists a unique critical mass $\lambda^{\rm HFB}=\lambda^{\rm HFB}(\kappa)>0$ of $I^{\rm HFB}_{m,\kappa,\theta}(\lambda)$, which is independent of $m$ and $\theta$, such that the following statements hold: Furthermore, the function $\lambda^{\rm HFB}(\kappa)$ is strictly decreasing, continuous with respect to $\kappa$, and s

Theorems & Definitions (32)

  • Proposition 1.1: Chandrasekhar Limit
  • Theorem 1.2: Existence in HFB theory
  • Theorem 1.3: Properties of HFB minimizers
  • Proposition 1.4: Gagliardo--Nirenberg inequality in HF theory CGNO-25
  • Theorem 1.5: Existence and nonexistence in HF theory
  • Remark 1.6: Blow-up profile
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Corollary 2.4
  • ...and 22 more