On one-dimensional Bose gases with two- and (critical) attractive three-body interactions
Dinh-Thi Nguyen, Julien Ricaud
TL;DR
This work analyzes a one-dimensional trapped Bose gas with both two-body and attractive three-body interactions, identifying stability regimes and deriving a comprehensive mean-field description via cubic-quintic NLS functionals. It establishes existence and collapse behavior of NLSNLS ground states, proves the Hartree ground states converge to NLS ground states, and extends these insights to the full many-body problem using quantum de Finetti theorems. The results distinguish critical and non-critical three-body regimes, delivering precise energy convergences and detailed condensation statements, including universal blow-up profiles described by the quintic NLS ground state when approaching criticality. The findings illuminate how two- and three-body interactions shape condensation and collapse in 1D, with explicit scaling laws and rigorous limits connecting microscopic quantum dynamics to effective nonlinear field theories.
Abstract
We consider a one-dimensional, trapped, focusing Bose gas where $N$ bosons interact with each other via both a two-body interaction potential of the form $a N^{α-1} U(N^α(x-y))$ and an attractive three-body interaction potential of the form $-b N^{2β-2} W(N^β(x-y,x-z))$, where $a\in\mathbb{R}$, $b,α>0$, $0<β<1$, $U, W \geq 0$, and $\int_{\mathbb{R}}U(x) \mathop{}\!\mathrm{d}x = 1 = \iint_{\mathbb{R}^2} W(x,y) \mathop{}\!\mathrm{d}x \mathop{}\!\mathrm{d}y$. The system is stable either for any $a\in\mathbb{R}$ as long as $b<\mathfrak{b} := 3π^2/2$ (the critical strength of the 1D focusing quintic nonlinear Schrödinger equation) or for $a \geq 0$ when $b=\mathfrak{b}$. In the former case, fixing $b \in (0,\mathfrak{b})$, we prove that in the mean-field limit the many-body system exhibits the Bose$\unicode{x2013}$Einstein condensation on the cubic-quintic NLS ground states. When assuming $b=b_N \nearrow \mathfrak{b}$ and $a=a_N \to 0$ as $N \to\infty$, with the former convergence being slow enough and "not faster" than the latter, we prove that the ground state of the system is fully condensed on the (unique) solution to the quintic NLS equation. In the latter case $b=\mathfrak{b}$ fixed, we obtain the convergence of many-body energy for small $β$ when $a > 0$ is fixed. Finally, we analyze the behavior of the many-body ground states when the convergence $b_N \nearrow \mathfrak{b}$ is "faster" than the slow enough convergence $0<a_N \searrow 0$.