Ground states of spin-1 BEC with attractive mean-field interaction trapped in harmonic potential in $${\\mathbb {R}}^2$$

Abstract

We investigate the ground states of spin-1 Bose–Einstein condensate trapped in harmonic potential in $${\\mathbb {R}}^2$$\n with attractive mean-field interaction constant $$c_0$$\n and spin-exchange interaction constant $$c_1$$\n , two conserved quantities, the number of atoms N and the total magnetization M are involved in. The existence and nonexistence of the ground states have been analyzed according to the relations among $$c_0$$\n , $$c_1$$\n and M. As $$(c_{0}$$\n , $$c_{1})$$\n approaches the thresholds, the asymptotic behavior of the ground states are presented in details, and the energy estimates, mass concentration and vanishing phenomena are described rigorously.