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Journal of Mathematical Physics

Existence and uniqueness of ground states for attractive Bose–Einstein condensates in box-shaped traps

Yan Li, Yong Luo

Published Mar 1, 2021• 4 citations
Publisher

Abstract

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).

DOI
10.1063/5.0039100

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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).
We study the Bose-Einstein condensates (BEC) in two or three dimensions with attractive interactions, described by L2L^{2}L2 constraint Gross-Pitaevskii energy functional. First, we give the precise description of the chemical potential of the condensate μμμ and the attractive interaction aaa. Next, for a class of degenerated trapping potential with non-isolated critical points, we obtain the existence and the local uniqueness of excited states by precise analysis of the concentrated points and the Lagrange multiplier. To our best knowledge, this is the first result concerning on excited states of BEC in Mathematics. Also, our results show that ka∗ka_*ka∗​ are critical values in two dimension when the concentration occurs for any positive integer kkk with some positive constant a∗a_*a∗​. And we point out that our results on degenerated trapping potential with non-isolated critical points are also new even for the classical Schrödinger equations. Here our main tools are finite-dimensional reduction and various Pohozave identities. The main difficulties come from the estimates on Lagrange multiplier and the different degenerate rate along different directions at the critical points of V(x)V(x)V(x).
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|
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 article is devoted to studying the model of two-dimensional attractive Bose-Einstein condensates in a trap V(x)V(x)V(x) rotating at the velocity $\Omega $. This model can be described by the complex-valued Gross-Pitaevskii energy functional. It is shown that there exists a critical rotational velocity 000 000 denotes the absolute product for the number of particles times the scattering length, and w>0w>0w>0 is the unique positive solution of Δw−w+w3=0\Delta w-w+w^3=0Δw−w+w3=0 in R2\mathbb{R}^2R2. If V(x)=∣x∣2V(x)=|x|^2V(x)=∣x∣2 and $ 0<\Omega <\Omega^*(=2)$ is fixed, we prove that, up to a constant phase, all minimizers must be real-valued, unique and free of vortices as a↗a∗a \nearrow a^*a↗a∗, by analyzing the refined limit behavior of minimizers and employing the non-degenerancy of www.
Abstract. We study ground states of two-dimensional Bose–Einstein condensates with repulsive ([Formula: see text]) or attractive ([Formula: see text]) interactions in a trap [Formula: see text] rotating at velocity [Formula: see text]. It is known that there exist critical parameters [Formula: see text] and [Formula: see text] such that if [Formula: see text], then there is no ground state for any [Formula: see text]; if [Formula: see text], then ground states exist if and only if [Formula: see text]. As a completion of the existing results, in this paper, we focus on the critical case where [Formula: see text] to classify the existence and nonexistence of ground states for any [Formula: see text]. Moreover, for a suitable class of radially symmetric traps [Formula: see text], employing the inductive symmetry method, we prove that up to a constant phase, ground states must be real valued, unique, and free of vortices as [Formula: see text], no matter whether the interactions of the condensates are repulsive or not.
In this paper, we mainly investigate complex-valued vortex states of trapped attractive Bose-Einstein condensates (BECs), which can be described by an $ L^2- $critical constraint minimization problem. We prove that both the existence and nonexistence of vortex states may occur at the threshold $ a^*(m) $ depending on the value of $ V(0) $. When there is no vortex states at the threshold $ a^*(m) $, the limit behavior of vortex states as $ a \nearrow a^*(m) $ is also investigated if the trapping potential $ V(x) $ vanishes on the unit circle $ B(0,1) $ in $ {{\mathbb{R}}}^2 $.