EXISTENCE AND ASYMPTOTIC BEHAVIOR OF GROUND STATES FOR ROTATING BOSE-EINSTEIN CONDENSATES

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 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 R; if 0 \leq \Omega < \Omega \ast , then ground states exist if and only if a \in (-a\ast , +oo ). As a completion of the existing results, in this paper, we focus on the critical case where 0 < \Omega = \Omega \ast < +oo to classify the existence and nonexistence of ground states for any 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, 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.

Automated summary

In this paper, we study the ground states of two-dimensional Bose-Einstein condensates with repulsive or attractive interactions in a rotating trap. We analyze the existence and nonexistence of ground states for different parameters and classify the critical case where the rotation velocity is equal to the critical velocity. Furthermore, for a specific class of traps, we prove that the ground states must be real valued, unique, and free of vortices as the rotation velocity approaches zero.