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 Ω . It is known that there exists a critical rotational velocity 0 < Ω ∗ : = Ω ∗ ( V ) ≤ ∞ and a critical number 0 < a ∗ < ∞ such that for any rotational velocity 0 ≤ Ω < Ω ∗ , 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 ↗ a ∗ , where Ω ∈ ( 0 , Ω ∗ ) is fixed.