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Kinetic localization via Poincaré-type inequalities and applications to the condensation of Bose gases

Jacky J. Chong, Hao Liang, Phan Thành Nam

TL;DR

The paper develops a simplified, self-contained localization framework for dilute Bose gases by proving a Poincaré-type inequality in finite-volume boxes and using Neumann-box localization to relate local condensation on subdomains to global Bose–Einstein condensation. The main technical advance is a generalized Poincaré inequality for cubes, which facilitates kinetic-energy localization and a cell-wise analysis that, combined with Bogoliubov theory, yields complete BEC for κ ∈ (0, 2/11) in the dilute regime. The approach unifies localization techniques with finite-volume spectral gaps to derive rigorous condensation results beyond the GP scaling, and it provides a streamlined path toward understanding BEC in intermediate regimes. The work positions a simpler, yet powerful, localization method as a versatile tool for BEC proofs, with potential extensions to more singular regimes and positive-temperature settings.

Abstract

We propose a simplified localization method for Bose gases, based on a Poincare-type inequality, which leads to a new derivation of Bose-Einstein condensation for dilute Bose gases beyond the Gross-Pitaevskii scaling regime.

Kinetic localization via Poincaré-type inequalities and applications to the condensation of Bose gases

TL;DR

The paper develops a simplified, self-contained localization framework for dilute Bose gases by proving a Poincaré-type inequality in finite-volume boxes and using Neumann-box localization to relate local condensation on subdomains to global Bose–Einstein condensation. The main technical advance is a generalized Poincaré inequality for cubes, which facilitates kinetic-energy localization and a cell-wise analysis that, combined with Bogoliubov theory, yields complete BEC for κ ∈ (0, 2/11) in the dilute regime. The approach unifies localization techniques with finite-volume spectral gaps to derive rigorous condensation results beyond the GP scaling, and it provides a streamlined path toward understanding BEC in intermediate regimes. The work positions a simpler, yet powerful, localization method as a versatile tool for BEC proofs, with potential extensions to more singular regimes and positive-temperature settings.

Abstract

We propose a simplified localization method for Bose gases, based on a Poincare-type inequality, which leads to a new derivation of Bose-Einstein condensation for dilute Bose gases beyond the Gross-Pitaevskii scaling regime.
Paper Structure (12 sections, 9 theorems, 103 equations)

This paper contains 12 sections, 9 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Generalized Poincaré inequality for cubes
  4. Bose--Einstein condensation of dilute Bose gases
  5. Remarks on previous results
  6. Poincaré-type Inequality
  7. Proof of Theorem \ref{['thm:BEC']}
  8. Kinetic energy localization
  9. Proof of Theorem \ref{['thm:BEC']}
  10. The GP Regime with Small Scattering Length
  11. Modified scattering function and Neumann symmetrization
  12. Proof of Proposition \ref{['prop:Neumann completing the square']}

Key Result

Theorem 1

Let $d\in \mathbb{N}$ and $p>1$. There exist a constant $C_{p,d}>0$, depending only on the dimension $d$ and $p$, such that holds for all $f\in W^{1,p}(\Lambda)$ and $M\in\mathbb{N}$. Consequently, for every $\varepsilon\in\left(0,\frac{C^{2}_{2,d}}{4\ell^{2}}\right]$ we have the Poincaré inequality in the sense of quadratic forms on $L^2(\Lambda)$. Here, $\Delta_{\Lambda}$ is the Neumann Laplac

Theorems & Definitions (20)

  • Theorem 1: Poincare inequality for cubes
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 2: Bose--Einstein condensation
  • Theorem 3
  • proof : Proof of Theorem \ref{['thm:Poincare']}
  • Lemma 1: Kinetic energy localization
  • proof
  • Lemma 2
  • ...and 10 more