科研之友
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摘要
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.
