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The Bose gas in a box with Neumann boundary conditions

Chiara Boccato, Robert Seiringer

Abstract

We consider a gas of bosonic particles confined in a box with Neumann boundary conditions. We prove Bose-Einstein condensation in the Gross-Pitaevskii regime, with an optimal bound on the condensate depletion. Our lower bound for the ground state energy in the box implies (via Neumann bracketing) a lower bound for the ground state energy of the Bose gas in the thermodynamic limit.

The Bose gas in a box with Neumann boundary conditions

Abstract

We consider a gas of bosonic particles confined in a box with Neumann boundary conditions. We prove Bose-Einstein condensation in the Gross-Pitaevskii regime, with an optimal bound on the condensate depletion. Our lower bound for the ground state energy in the box implies (via Neumann bracketing) a lower bound for the ground state energy of the Bose gas in the thermodynamic limit.
Paper Structure (14 sections, 16 theorems, 356 equations)

This paper contains 14 sections, 16 theorems, 356 equations.

Table of Contents

  1. Introduction and main results
  2. Excitation Hamiltonians
  3. Analysis of the excitation Hamiltonian
  4. Generalized Bogoliubov transformation
  5. Analysis of $\mathcal{G}_{n,\ell}^{(0)}$
  6. Analysis of $\mathcal{G}_{n,\ell}^{(1)}$
  7. Analysis of $\mathcal{G}_{n,\ell}^{(2)}$
  8. Analysis of $e^{-B}\mathcal{K} e^B$
  9. Analysis of $e^{-B}\mathcal{L}_{n,\ell}^{(2,V)}e^B$
  10. Analysis of $\mathcal{G}_{n,\ell}^{(3)}$
  11. Analysis of $\mathcal{G}_{n,\ell}^{(4)}$
  12. Proof of Proposition \ref{['prop:G']}
  13. Proof of Theorem \ref{['mainTheorem']} and Corollary \ref{['LHY']}
  14. The two-body problem in the Neumann box

Key Result

Theorem 1.1

Let $V$ be positive, compactly supported, spherically symmetric and bounded, and assume that $\kappa$ is a fixed, small enough constant independent of all parameters and $n\ell^{-1}\leq 1$. Then, the ground state energy $e_{n,\ell}$ of $H_{n,\ell}$ defined in eq:H1 is such that for a constant $C>0$ depending only on V. Furthermore, let $\psi_n \in L^2_s (\Lambda_1^n)$ be a normalized wave functio

Theorems & Definitions (30)

  • Theorem 1.1
  • Corollary 1.2
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Lemma 3.1
  • Lemma 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • ...and 20 more