A separation threshold for ground states of two-component attractive Bose--Einstein condensates in displaced harmonic traps

Abstract

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 $x_1\ne x_2\in {\mathbb{R}}^2$. For intraspecies interactions $a_1,a_2\in (0,a^{\ast })$, 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 $|x_1-x_2|\ge d_c$. The threshold is given by $d_c={\Lambda }_{\ast }^{-1/4}$, where ${\Lambda }_{\ast }$ 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.