具有二体和三体相互作用的旋转二维玻色气体平均场极限中的凝聚与坍缩

中文摘要 AI 辅助

我们考虑在 $\mathbb{R}^2$ 中旋转谐振陷阱内的 $N$ 个相互作用玻色子系统，其中二体相互作用是吸引性的，尺度为 $N^{2\alpha}U(N^\alpha x)$，其中 $0<\alpha<1/12$；三体相互作用是排斥性的，尺度为 $N^{4\beta}W(N^\beta x,N^\beta y)$，其中 $0<\beta<1/24$。在平均场极限下，基态能量有效地由旋转三次-五次非线性薛定谔泛函描述。我们分析了坍缩区域，其中二体耦合 $a$ 接近临界值 $a_*$，三体耦合 $b$ 趋于零。NLS 基态以由 Gagliardo-Nirenberg 不等式的最优函数给出的普适轮廓发生爆炸，且能量满足 $E^{\mathrm{NLS}} = (1-\zeta/4+o(1))\mathcal{Q}{\rm{pot}}\ell_n^2$，其中 $\zeta\geqslant0$ 由 $a_n\to a_*$ 和 $b_n\searrow0$ 的相对速率决定。从多体理论出发，我们严格证明了这一有效描述：在坍缩区域中，量子基态能量收敛到 NLS 能量，且具有相同的渐近展开；多体基态表现出对普适爆炸轮廓的完全玻色-爱因斯坦凝聚。

英文摘要

We consider a system of $N$ interacting bosons in a rotating harmonic trap in $\mathbb{R}^2$, where the two-body interaction is attractive and scaled as $N^{2α}U(N^αx)$ with $0<α<1/12$, and the three-body interaction is repulsive and scaled as $N^{4β}W(N^βx,N^βy)$ with $0<β<1/24$. In the mean-field limit, the ground state energy is effectively described by a rotating cubic-quintic nonlinear Schrödinger functional. We analyze the collapse regime where the two-body coupling $a$ approaches the critical value $a_*$ and the three-body coupling $b$ tends to zero. The NLS ground states blow up with a universal profile given by the optimizer of the Gagliardo-Nirenberg inequality, and the energy satisfies $E^{\mathrm{NLS}} = (1-ζ/4+o(1))\mathcal{Q}{\rm{pot}}\ell_n^2$ with $ζ\geqslant0$ determined by the relative rates of $a_n\to a_*$ and $b_n\searrow0$. From the many-body theory, we rigorously justify this effective description: the quantum ground state energy converges to the NLS energy with the same asymptotic expansion in the collapse regime, and the many-body ground states exhibit complete Bose-Einstein condensation onto the universal blow-up profile.

发表机构

• School of Mathematics and Statistics, Lanzhou University（兰州大学数学与统计学院）
• School of Mathematical Sciences, Zhejiang Normal University（浙江师范大学数学系）

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