Local Uniqueness of Ground States for Rotating Bose-Einstein Condensates with Attractive Interactions

We study ground states of two-dimensional Bose-Einstein condensates with attractive interactions in a trap $V(x)$ rotating at the velocity $\Omega$. It is known that there exist a critical rotational velocity $0<\Omega ^*:=\Omega^*(V)\leq \infty$ and a critical number $0<a^*<\infty$ such that for any rotational velocity $0\le \Omega <\Omega ^*$, ground states exist if and only if the coupling constant $a$ satisfies $a<a^*$. For a general class of traps $V(x)$, which may not be symmetric, we prove in this paper that up to a constant phase, there exists a unique ground state as $a\nearrow a^*$, where $\Omega\in(0,\Omega^*)$ is fixed. This result extends essentially our recent uniqueness result, where only the radially symmetric traps $V(x)$ could be handled with.