We study the collapse of a many-body system which is used to model two-component Bose–Einstein condensates with attractive intra-species interactions and either attractive or repulsive inter-species interactions. Such a system consists a mixture of two different species for N identical bosons in \({\mathbb {R}}^2\), interacting with potentials rescaled in the mean-field manner \(-N^{2\beta -1}w^{(\sigma )}(N^{\beta }x)\) with \(\int \limits _{{\mathbb {R}}^{2}}w^{(\sigma )}(x)\mathrm{d}x=1\). Assuming that \(0<\beta <1/2\), we first show that the leading order of the quantum energy is captured correctly by the Gross–Pitaevskii energy. Secondly, we investigate the blow-up behavior of the quantum energy as well as the ground states when \(N\rightarrow \infty \), and either the total interaction strength of intra-species and inter-species or the strengths of intra-species interactions of each component approach sufficiently slowly a critical value, which is the critical strength for the focusing Gross–Pitaevskii functional. We prove that the many-body ground states fully condensate on the (unique) Gagliardo–Nirenberg solution.

我们研究了多体系统的崩溃，该系统用于模拟具有吸引力的种内相互作用以及吸引力或排斥的种间相互作用的两组分玻色-爱因斯坦冷凝物。这样的系统由\（{\ mathbb {R}} ^ 2 \）中N个相同玻色子的两种不同物质组成的混合物，与以均场方式\（-N ^ {2 \ beta -1 } w ^ {（\ sigma}}（N ^ {\ beta} x）\）与\（\ int \ limits _ {{\ mathbb {R}} ^ {2}} w ^ {（\ sigma}}（ x）\ mathrm {d} x = 1 \）。假设\（0 <\ beta <1/2 \），我们首先证明，Gross–Pitaevskii能量正确地捕获了量子能量的前导顺序。其次，我们研究量子能量的爆炸行为以及\（N \ rightarrow \ infty \）时的基态，以及种间和种间的总相互作用强度或种内强度。每个组成部分的物种相互作用都足够缓慢地接近临界值，这是聚焦Gross-Pitaevskii功能的临界强度。我们证明了多体基态在（唯一的）Gagliardo-Nirenberg解上完全凝聚。