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Mass Concentration of Two-Spinless Fermi Systems with Attractive Interactions

Yujin Guo, Yan Li

Abstract

We study the two-spinless mass-critical Fermi systems with attractive interactions and trapping potentials. We prove that ground states of the system exist, if and only if the strength $a$ of attractive interactions satisfies $0<a<a_2^*$, where $0<a_2^*<+\infty$ is the best constant of a dual finite-rank Lieb-Thirring inequality. By the blow-up analysis of many-fermion systems, we show that ground states of the system concentrate at the flattest minimum points of the trapping potential $V(x)$ as $a\nearrow a_2^*$.

Mass Concentration of Two-Spinless Fermi Systems with Attractive Interactions

Abstract

We study the two-spinless mass-critical Fermi systems with attractive interactions and trapping potentials. We prove that ground states of the system exist, if and only if the strength of attractive interactions satisfies , where is the best constant of a dual finite-rank Lieb-Thirring inequality. By the blow-up analysis of many-fermion systems, we show that ground states of the system concentrate at the flattest minimum points of the trapping potential as .
Paper Structure (5 sections, 9 theorems, 162 equations)

This paper contains 5 sections, 9 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Existence and Non-existence of Minimizers
  3. Estimates of Minimizers as $a\nearrow a_2^*$
  4. Mass Concentration of Minimizers as $a\nearrow a_2^*$
  5. Appendix

Key Result

Theorem 1.1

Let $a_2^*>0$ be defined by 0.7, and assume the potential $0\le V(x)\in L_{loc}^\infty({\mathbb R}^3)$ satisfies $\lim_{|x|\to\infty}V(x)=\infty$. Then we have

Theorems & Definitions (11)

  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Lemma 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 4.1
  • Lemma 4.2
  • ...and 1 more