Existence and Asymptotic Behavior of Ground States for Rotating Bose-Einstein Condensates

Abstract

We study ground states of two-dimensional Bose-Einstein condensates with repulsive ($a>0$) or attractive ($a<0$) interactions in a trap $V(x)$ rotating at the velocity $\Omega$. It is known that there exist critical parameters $a^{\ast }>0$ and ${\Omega }^{\ast }:={\Omega }^{\ast }(V(x))>0$ such that if $\Omega >{\Omega }^{\ast }$, then there is no ground state for any $a\in \mathbb{R}$; if $0\le \Omega <{\Omega }^{\ast }$, then ground states exist if and only if $a\in (-a^{\ast },+\infty )$. As a completion of the existing results, in this paper, we focus on the critical case where $0<\Omega ={\Omega }^{\ast }<+\infty$ and classify the existence and nonexistence of ground states for $a\in \mathbb{R}$. Moreover, for a suitable class of radially symmetric traps $V(x)$, employing the inductive symmetry method, we prove that up to a constant phase, the ground states must be real-valued, unique and free of vortices as $\Omega \searrow 0$, no matter whether the interactions of the condensates are repulsive or not.

arXiv comment33 pages

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