Free and Harmonic Trapped Spin-1 Bose–Einstein Condensates in [math]

SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4375-4414, August 2024.
Abstract. We investigate physical states of spin-1 Bose–Einstein condensate in [math] with mean-field interaction constant [math] and spin-exchange interaction constant [math], two conserved quantities, the number of atoms [math], and the total magnetization [math] are involved in. First, in the free case, existence and asymptotic behavior of ground states are analyzed according to the relations among [math], [math], [math], and [math]. Furthermore, we show that the corresponding standing wave is strongly unstable. When the atoms are trapped in a harmonic potential, we prove the existence of ground states and excited states along with some precisely asymptotics. Besides, we get that the set of ground states is stable under the associated Cauchy flow while the excited state corresponds to a strongly unstable standing wave. Our results not only show some characteristics of three-dimensional spin-1 BEC under the effect between the spin-dependent interaction and the external magnetic field, but also support some experimental observations as well as numerical results on spin-1 BEC.