On one-dimensional Bose gases with two- and (critical) attractive three-body interactions

TL;DR

This paper studies a one-dimensional Bose gas with combined two- and three-body interactions, analyzing stability, ground state condensation, and energy convergence in various regimes approaching critical interaction strength.

Contribution

It provides a rigorous analysis of the stability and condensation phenomena in a 1D Bose gas with critical attractive three-body interactions, extending understanding of mean-field limits and nonlinear Schrödinger equations.

Findings

System is stable for any a if b<critical value or for a≥0 when b equals critical value.

Ground states exhibit Bose–Einstein condensation on cubic-quintic NLS ground states in certain regimes.

Energy convergence results are established near the critical interaction strength.

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 $a N^{\alpha-1} U(N^\alpha(x-y))$ and an attractive three-body interaction potential of the form $-b N^{2\beta-2} W(N^\beta(x-y,x-z))$, where $a\in\mathbb{R}$, $b,\alpha>0$, $0<\beta<1$, $U, W \geq 0$, 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 $b<\mathfrak{b} := 3\pi^2/2$ (the critical strength of the 1D focusing quintic nonlinear Schr\"odinger 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 $\beta$ 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 $0<a_N \searrow 0$.