发表机构
Chongqing University of Arts and Sciences; Southwest University（重庆文理学院; 西南大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究位移谐振子阱中双组分吸引性玻色-爱因斯坦凝聚体，证明在临界耦合下存在分离阈值，低于阈值无基态，高于阈值有基态，并给出阈值表达式，推广了已有分类结果。
AI 中文摘要
本文研究具有位移谐振子囚禁势 $V_1(x)=|x-x_1|^2$ 和 $V_2(x)=|x-x_2|^2$（其中 $x_1\neq x_2\in\mathbb R^2$）的双组分吸引性玻色-爱因斯坦凝聚体基态的存在性。对于组分内相互作用 $a_1,a_2\in (0,a^*)$，我们关注临界组分间耦合强度 $\beta=\beta^*:=a^*+\sqrt{(a^*-a_1)(a^*-a_2)}$，并证明存在一个分离阈值 $d_c\in(0,\infty)$，使得当 $0<|x_1-x_2|<d_c$ 时相应的约束极小化问题没有极小元，而当 $|x_1-x_2|\geq d_c$ 时存在极小元。该阈值由 $d_c=\Lambda_*^{-1/4}$ 给出，其中 $\Lambda_*$ 由一个辅助变分问题刻画且可达。我们的结果推广了 Guo 等人 [J. Funct. Anal. 276 (2019)] 的先前工作，并完成了具有谐振子囚禁势的双组分吸引性玻色-爱因斯坦凝聚体基态存在性的完整分类。
英文摘要
In this paper, we study the existence of ground states for two-component attractive Bose--Einstein condensates with displaced harmonic trapping potentials \[ V_1(x)=|x-x_1|^2 \ \text{and}\ V_2(x)=|x-x_2|^2, \] where $x_1\neq x_2\in\mathbb R^2$. For intraspecies interactions $a_1,a_2\in (0,a^*)$, we focus on the critical interspecies coupling \[ β=β^*:=a^*+\sqrt{(a^*-a_1)(a^*-a_2)} \] and prove that there exists a separation threshold $d_c\in(0,\infty)$ such that the corresponding constrained minimization problem admits no minimizer when $0<|x_1-x_2|<d_c$, whereas it admits a minimizer when $|x_1-x_2|\geq d_c$. The threshold is given by $d_c=Λ_*^{-1/4}$, where $Λ_*$ is characterized by an auxiliary variational problem and attained. Our results extend the previous work of Guo et al. [J. Funct. Anal. 276 (2019)] and complete the existence classification of ground states for two-component attractive Bose--Einstein condensates with harmonic trapping potentials.
