Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Derivation of the 3D quintic Gross--Pitaevskii equation

Arnaud Triay, François L. A. Visconti

TL;DR

This work rigorously derives the time-dependent quintic Gross–Pitaevskii equation in three dimensions from the many-body Schrödinger dynamics of bosons with a three-body interaction in the Gross–Pitaevskii scaling, proving propagation of Bose–Einstein condensation in time. The authors introduce a multi-scale localisation to handle short-range three-body correlations, apply a cubic Bogoliubov transform to remove correlations, and develop renormalised fluctuation operators whose dynamics satisfy a Grönwall-type bound. They establish universality of the coupling constant, showing it is determined by the three-body scattering hypervolume $b(V)$ and not by microscopic details, and prove quantitative convergence of the reduced density matrices to the condensate factorization $\varphi_t^{\otimes k}$. The results extend the BBGKY-type program from two-body to three-body interactions, providing sharp stability and fluctuation bounds with an $O(N^{-1/2})$ convergence rate (up to exponential-in-time factors) in the GP regime, with potential relevance for experiments probing three-body effects in Bose gases.

Abstract

We study the time evolution of Bose--Einstein condensates with three-body interactions in the Gross--Pitaevskii regime. We show that Bose--Einstein condensation is preserved under many-body evolution and that the condensate wavefunction evolves according to the quintic Gross--Pitaevskii equation in $\mathbb{R}^3$, which is energy critical. In particular, we show that the effective coupling constant is universal and depends only on a three-body scattering hypervolume.

Derivation of the 3D quintic Gross--Pitaevskii equation

TL;DR

This work rigorously derives the time-dependent quintic Gross–Pitaevskii equation in three dimensions from the many-body Schrödinger dynamics of bosons with a three-body interaction in the Gross–Pitaevskii scaling, proving propagation of Bose–Einstein condensation in time. The authors introduce a multi-scale localisation to handle short-range three-body correlations, apply a cubic Bogoliubov transform to remove correlations, and develop renormalised fluctuation operators whose dynamics satisfy a Grönwall-type bound. They establish universality of the coupling constant, showing it is determined by the three-body scattering hypervolume and not by microscopic details, and prove quantitative convergence of the reduced density matrices to the condensate factorization . The results extend the BBGKY-type program from two-body to three-body interactions, providing sharp stability and fluctuation bounds with an convergence rate (up to exponential-in-time factors) in the GP regime, with potential relevance for experiments probing three-body effects in Bose gases.

Abstract

We study the time evolution of Bose--Einstein condensates with three-body interactions in the Gross--Pitaevskii regime. We show that Bose--Einstein condensation is preserved under many-body evolution and that the condensate wavefunction evolves according to the quintic Gross--Pitaevskii equation in , which is energy critical. In particular, we show that the effective coupling constant is universal and depends only on a three-body scattering hypervolume.
Paper Structure (37 sections, 22 theorems, 494 equations, 2 figures)

This paper contains 37 sections, 22 theorems, 494 equations, 2 figures.

Table of Contents

  1. Introduction and main result
  2. The model
  3. Main result
  4. Notation, strategy and proof of Theorem \ref{['th:main_result']}
  5. Notation and preliminaries
  6. Factoring out the condensate
  7. Fluctuations around the condensate.
  8. Removing correlations
  9. Approximating the cubic transformation.
  10. Proof of Theorem \ref{['th:main_result']}
  11. Analysis of fluctuations
  12. Three-body scattering solution
  13. Analysis of $\mathcal{N}^\textrm{\normalfont ren}$ and $\mathcal{Q}^\textrm{\normalfont ren}$: proof of Lemma \ref{['lemma:number_operator_renormalised']}
  14. Analysis of the generator of the fluctuations
  15. Proof of Lemma \ref{['lemma:generator_estimate']}
  16. ...and 22 more sections

Key Result

Theorem 1.2

Let $V$ satisfy Assumption ass:potential. Let $(\Psi_{N,0})_N$ be a sequence of normalised wavefunctions in $\mathfrak{H}^N$, and let $\gamma_N^{(1)}$ denote the $1$-particle reduced density matrix of $\Psi_{N,0}$. Let $\varphi_0\in H^6(\mathbb{R}^3)$ be normalised and define and Let $\Psi_{N,t} = e^{-\mathbf{i} tH_N}\Psi_{N,0}$ be the solution to the Schrödinger equation eq:schroedinger_eq_time

Figures (2)

  • Figure 1: Illustration of a coverage of the box ${\color{Red}\Lambda_r^{(\ell_1)}}$ by the smaller boxes ${\color{Gray}\Lambda_s^{(\ell_3)}}$. Some of the grey boxes overlap only partially with the red one. The quantities $M_{r,s}^{(\ell_1,\ell_3)}$ and $M_{r,t}^{(\ell_1,\ell_3)}$ respectively count the number of particles in the green and blue rectangles.
  • Figure 2: Illustration of the boxes ${\color{Blue}\Lambda_r^{(\ell_1)}}$ and ${\color{Red}\Lambda_r^{(3\ell_1)}}$. Because the interaction has range $\ell_2$, which is less than $\ell_1$, particles located in the box $\Lambda_r^{(\ell_1)}$ can only interact with particles located in the box $\Lambda_r^{(3\ell_1)}$.

Theorems & Definitions (35)

  • Theorem 1.2
  • Proposition 2.1: Main Grönwall estimate
  • Lemma 2.2: Properties of $\mathcal{N}^\textrm{\normalfont ren}$ and $\mathcal{Q}^\textrm{\normalfont ren}$
  • Lemma 3.1: Properties of $\omega_{\lambda,N}, \varepsilon_\lambda$ and $u_\lambda$
  • proof : Proof of Lemma \ref{['lemma:truncated_three_body_scattering_solution']}
  • Lemma 3.2
  • Lemma 3.3
  • Proposition 3.4: Estimate on effective two-body potentials
  • Proposition 3.5: Further estimates on effective interaction potentials
  • Remark 3.6
  • ...and 25 more