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arXiv:2609.27316

Critical surface for two component Bose-Einstein Condensates

Peng Wenshuai, Xiaoyu Zeng, Qidi Zhang, Huan‐Song Zhou

Published Sep 23, 2026• 0 citations
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Abstract

We investigate the existence of ground states for two-component Bose--Einstein condensates with intraspecies interactions $a_1, a_2 \in (0, a^*)$ and interspecies interaction $β>0$. By carefully investigating an associated auxiliary minimization problem, we prove the existence of a unique, continuous, critical surface $γ= γ(a_1, a_2)$ between $β_*:= \sqrt{(a^* - a_1) (a^* - a_2)}$ and $β^*:= a^* - \frac{a_1 + a_2}{2}$. This surface provides a complete classification for the existence of minimizers: ground states exist for $β\in (0,γ)$ and do not exist for $β> γ$. Additionally, when the trapping potentials are continuous at a common zero point, there is no ground state for $β= γ$. Our results close the gap case $β\in [β_*, β^*]$ when $a_1\ne a_2$ in literature, significantly extending the results in \cite{Bao-Cai-1,guoBlowupSolutionsTwo2017a}.

DOI
10.48550/arxiv.2609.27316

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In this paper, we study the existence of ground states for two-component attractive Bose--Einstein condensates with displaced harmonic trapping potentials V1(x)=∣x−x1∣2 and V2(x)=∣x−x2∣2, V_1(x)=|x-x_1|^2 \ \text{and}\ V_2(x)=|x-x_2|^2, V1​(x)=∣x−x1​∣2 and V2​(x)=∣x−x2​∣2, where x1≠x2∈R2x_1\neq x_2\in\mathbb R^2x1​=x2​∈R2. For intraspecies interactions a1,a2∈(0,a∗)a_1,a_2\in (0,a^*)a1​,a2​∈(0,a∗), we focus on the critical interspecies coupling β=β∗:=a∗+(a∗−a1)(a∗−a2) β=β^*:=a^*+\sqrt{(a^*-a_1)(a^*-a_2)} β=β∗:=a∗+(a∗−a1​)(a∗−a2​)​ and prove that there exists a separation threshold dc∈(0,∞)d_c\in(0,\infty)dc​∈(0,∞) such that the corresponding constrained minimization problem admits no minimizer when $0<|x_1-x_2|
As a continuation of our prior article, we study new pattern formations of ground states (u1,u2)(u_1,u_2)(u1​,u2​) for two-component Bose-Einstein condensates (BEC) with homogeneous trapping potentials in R2\mathbb {R}^2R2, where the intraspecies interaction (−a,−b)(-a,-b)(−a,−b) and the interspecies interaction −β-\beta−β are both attractive, i.e., aaa, bbb, and β\betaβ are all positive. If 0<b<a∗:=∥w∥220<b<a^*:=\|w\|^2_20<b<a∗:=∥w∥22​ and 0<β<a∗0<\beta <a^*0<β<a∗ are fixed, where www is the unique positive solution of Δw−w+w3=0\Delta w-w+w^3=0Δw−w+w3=0 in R2\mathbb {R}^2R2, the semi-trivial behavior of (u1,u2)(u_1,u_2)(u1​,u2​) as a↗a∗a\nearrow a^*a↗a∗ is proved in the sense that u1u_1u1​ concentrates at a unique point and while u2≡0u_2\equiv 0u2​≡0 in R2\mathbb {R}^2R2. However, if 0<b<a∗0<b<a^*0<b<a∗ and a∗≤β<β∗=a∗+(a∗−a)(a∗−b)a^*\le \beta <\beta ^*=a^*+\sqrt {(a^*-a)(a^*-b)}a∗≤β<β∗=a∗+(a∗−a)(a∗−b)​, the refined spike profile and the uniqueness of (u1,u2)(u_1,u_2)(u1​,u2​) as a↗a∗a\nearrow a^*a↗a∗ are analyzed, where (u1,u2)(u_1,u_2)(u1​,u2​) must be unique, u1u_1u1​ concentrates at a unique point, and meanwhile u2u_2u2​ can either blow up or vanish, depending on how β\betaβ approaches a∗a^*a∗.
This paper is concerned with two-component Bose–Einstein condensates with both attractive intraspecies and interspecies interactions passing an obstacle in a plane, which can be described by the ground states of the nonlinear Schrödinger system defined in an exterior domain Ω=R2\ω, with ω⊂R2 being a bounded smooth convex domain. Under the assumption that the trapping potentials Vi(x) for i = 1, 2 attain their global minima only on the whole boundary ∂Ω, the existence, non-existence, and limiting behavior of ground states for the system are studied. When intraspecies interactions a1 and a2 satisfy 0 < a1, a2 < a* and interspecies interaction β satisfies 0 < β < β* by the delicate energy analysis, an optimal blow-up rate for ground states is also given as β ↗ β*, where β*=a*+(a*−a1)(a*−a2), a*≔‖Q‖22, and Q is the unique positive solution of ΔQ − Q + Q3 = 0 in R2.
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