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The Lee-Huang-Yang energy for a dilute gas of hard spheres: an upper bound

Giulia Basti, Morris Brooks, Serena Cenatiempo, Alessandro Olgiati, Benjamin Schlein

Abstract

We consider a quantum gas consisting of $N$ hard spheres with radius $\frak{a} > 0$, obeying bosonic statistics and moving in the box $Λ= [0;L]^3$ with periodic boundary conditions. We are interested in the ground state energy per unit volume in the thermodynamic limit, with $N, L \to \infty$ at fixed density $ρ= N / L^3$. We derive an upper bound for the ground state energy density, matching the famous Lee-Huang-Yang formula, up to lower order terms, in the dilute limit $ρ\frak{a}^3 \ll 1$.

The Lee-Huang-Yang energy for a dilute gas of hard spheres: an upper bound

Abstract

We consider a quantum gas consisting of hard spheres with radius , obeying bosonic statistics and moving in the box with periodic boundary conditions. We are interested in the ground state energy per unit volume in the thermodynamic limit, with at fixed density . We derive an upper bound for the ground state energy density, matching the famous Lee-Huang-Yang formula, up to lower order terms, in the dilute limit .
Paper Structure (12 sections, 20 theorems, 545 equations)

This paper contains 12 sections, 20 theorems, 545 equations.

Table of Contents

  1. Introduction
  2. The Trial State
  3. Proof of Theorem \ref{['thm:main']}
  4. A-priori bounds on trial state
  5. Computation of the energy
  6. Proof of Proposition \ref{['prop:1-1']}
  7. Proof of Proposition \ref{['prop:2-2']}
  8. Proof of Proposition \ref{['prop:3-3']}
  9. Proof of Proposition \ref{['prop:1-2']}
  10. Proof of Proposition \ref{['prop:1-3']}
  11. Proof of Proposition \ref{['prop:2-3']}
  12. Proof of Lemma \ref{['lm:eta']}

Key Result

Theorem 1.1

For $\rho > 0$, let $e(\rho)$ be defined as in (def:e-rho). Then there exists $C > 0$ and a sufficiently small $\delta > 0$ such that for all $\rho > 0$ small enough.

Theorems & Definitions (32)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • Proposition 2.5
  • Lemma 3.1
  • proof
  • ...and 22 more