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

TL;DR

This paper investigates the existence, nonexistence, and properties of ground states in rotating Bose-Einstein condensates, especially at critical rotation speeds, revealing conditions for their uniqueness, reality, and vortex absence.

Contribution

It completes the classification of ground state existence at the critical rotation speed and proves ground state properties such as real-valuedness and vortex-free nature for certain traps.

Findings

Ground states exist for certain interaction strengths below a critical rotation speed.

At the critical rotation speed, the paper classifies when ground states exist or not.

Ground states are shown to be real-valued, unique, and vortex-free as rotation speed approaches zero.

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 $a^*>0$ and $\Omega ^*:=\Omega^*(V(x))>0$ such that if $\Omega>\Omega^*$, then there is no ground state for any $a\in\R$; if $0\le \Omega <\Omega ^*$, then ground states exist if and only if $a\in(-a^*,+\infty)$. As a completion of the existing results, in this paper, we focus on the critical case where $0<\Omega=\Omega^*<+\infty$ and classify the existence and nonexistence of ground states for $a\in\R$. 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.