FREE AND HARMONIC TRAPPED SPIN-1 BOSE--EINSTEIN CONDENSATES IN R3*

We investigate physical states of spin-1 Bose--Einstein condensate in R3 with meanfield interaction constant c0 and spin-exchange interaction constant c1, two conserved quantities, the number of atoms N, and the total magnetization M are involved in. First, in the free case, existence and asymptotic behavior of ground states are analyzed according to the relations among c0, c1, N, and M. 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.