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Ground state energy of dense gases of strongly interacting fermions

Søren Fournais, Błażej Ruba, Jan Philip Solovej

Abstract

We study the ground state energy of a gas of $N$ fermions confined to a unit box in $d$ dimensions. The particles interact through a 2-body potential with strength scaled in an $N$-dependent way as $N^{-α}v$, where $α\in \mathbb R$ and $v$ is a function of positive type satisfying a mild regularity assumption. Our focus is on the strongly interacting case $α<1-\frac2d$. We contrast our result with existing results in the weakly interacting case $α>1-\frac2d$, and the transition happening at the mean-field scaling $α=1-\frac2d$. Our proof is an adaptation of the bosonization technique used to treat the mean-field case.

Ground state energy of dense gases of strongly interacting fermions

Abstract

We study the ground state energy of a gas of fermions confined to a unit box in dimensions. The particles interact through a 2-body potential with strength scaled in an -dependent way as , where and is a function of positive type satisfying a mild regularity assumption. Our focus is on the strongly interacting case . We contrast our result with existing results in the weakly interacting case , and the transition happening at the mean-field scaling . Our proof is an adaptation of the bosonization technique used to treat the mean-field case.
Paper Structure (3 sections, 7 theorems, 70 equations)

This paper contains 3 sections, 7 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Statement of the main result
  3. Proof

Key Result

Proposition 1

Suppose that the Fourier transform $\widehat{v} : (2 \pi \mathbb{Z})^d \to \mathbb{R}$ of $v$ satisfies Introduce the quantity The minimum $E_N$ of the spectrum of the operator eq:HN_def satisfies for some positive constant $c$.

Theorems & Definitions (13)

  • Proposition 1
  • Theorem 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • ...and 3 more