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arXiv:2610.05183

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

Shubin Yu, Chunlei Tang

Published Oct 4, 2026• 0 citations
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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 $x_1\neq x_2\in\mathbb R^2$. 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|

DOI
10.48550/arxiv.2610.05183

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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 [ V_1(x)=|x-x_1|^2 \ \text{and}\ V_2(x)=|x-x_2|^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^+\sqrt{(a^-a_1)(a^-a_2)} ] 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|
We investigate the existence of ground states for two-component Bose--Einstein condensates with intraspecies interactions a1,a2∈(0,a∗)a_1, a_2 \in (0, a^*)a1​,a2​∈(0,a∗) and interspecies interaction β>0β>0β>0. By carefully investigating an associated auxiliary minimization problem, we prove the existence of a unique, continuous, critical surface γ=γ(a1,a2)γ= γ(a_1, a_2)γ=γ(a1​,a2​) between β∗:=(a∗−a1)(a∗−a2)β_*:= \sqrt{(a^* - a_1) (a^* - a_2)}β∗​:=(a∗−a1​)(a∗−a2​)​ and β∗:=a∗−a1+a22β^*:= a^* - \frac{a_1 + a_2}{2}β∗:=a∗−2a1​+a2​​. This surface provides a complete classification for the existence of minimizers: ground states exist for β∈(0,γ)β\in (0,γ)β∈(0,γ) and do not exist for β>γβ> γβ>γ. Additionally, when the trapping potentials are continuous at a common zero point, there is no ground state for β=γβ= γβ=γ. Our results close the gap case β∈[β∗,β∗]β\in [β_*, β^*]β∈[β∗​,β∗] when a1≠a2a_1\ne a_2a1​=a2​ in literature, significantly extending the results in \cite{Bao-Cai-1,guoBlowupSolutionsTwo2017a}.
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.
This paper is concerned with ground states of two-component Bose gases confined in a harmonic trap V(x)=x12+Λ2x22V(x)=x_1^2+Λ^2 x_2^2V(x)=x12​+Λ2x22​ rotating at the velocity Ω>0Ω>0Ω>0, where Λ≥1Λ\ge 1Λ≥1 and (x1,x2)∈R2(x_1, x_2)\in R^2(x1​,x2​)∈R2. We focus on the case where the intraspecies interaction (−a1,−a2)(-a_1,-a_2)(−a1​,−a2​) and the interspecies interaction −β-β−β are both attractive, i.e, a1,a2a_1, a_2a1​,a2​ and βββ are all positive. It is shown that for any 0<Ω<Ω∗:=20<Ω<Ω^*:=20<Ω<Ω∗:=2, ground states exist if and only if 000000 is the unique positive solution of −Δw+w−w3=0-Δw+ w-w^3=0−Δw+w−w3=0 in R2R^2R2. By developing the argument of refined expansions, we further prove the nonexistence of vortices for ground states as β↗β∗β\nearrowβ^*β↗β∗, where 0<Ω<Ω∗0<Ω<Ω^*0<Ω<Ω∗ and $0
We consider ground states of two-dimensional Bose–Einstein condensates in box-shaped trapping potentials Vext(x) with inhomogeneous attractive interactions am(x), which can be described equivalently by minimizers of Gross–Pitaevskii energy functional in bounded domains. In this paper, we prove that there is a threshold a* > 0 such that minimizers exist for 0 < a < a* and the minimizer does not exist for any a > a*. However, if a = a*, it is shown that whether minimizers exist depends sensitively on the asymptotic behaviors of m(x) near its maximum points. Moreover, based on a detailed analysis on the limit behavior of minimizers as a ↗ a*, we prove local uniqueness of minimizers under some suitable assumptions on m(x).