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

In this paper, we study the existence of ground states for two-component attractive Bose–Einstein condensates with displaced harmonic trapping potentials $$V_1(x)=\mathrm{\mid }x-x_1{\mathrm{\mid }}^2\text{ and }{V}_2(x)=\mathrm{\mid }x-x_2{\mathrm{\mid }}^2,$$ where $x_1\mathrm{\ne }{x}_2\in {\mathbb{R}}^2$. For intraspecies interactions $a_1,a_2\in (0,a^{\ast })$, we focus on the critical interspecies coupling $$\beta ={\beta }^{\ast }:=a^{\ast }+\sqrt{(a^{\ast }-a_1)(a^{\ast }-a_2)}$$ and prove that there exists a separation threshold $d_c\in (0,\mathrm{\infty })$ such that the corresponding constrained minimization problem admits no minimizer when $0<\mathrm{\mid }{x}_1-x_2\mathrm{\mid }<d_c$, whereas it admits a minimizer when $\mathrm{\mid }{x}_1-x_2\mathrm{\mid }\ge d_c$. The threshold is given by $d_c={\mathrm{\Lambda }}_{\ast }^{-1\mathrm{/}4}$, where ${\mathrm{\Lambda }}_{\ast }$ 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.