We investigate physical states of spin-2 Bose–Einstein condensate in \({\mathbb {R}}^{d}(d=1,2,3)\) in terms of spin-independent interaction \(\tau \) , spin-exchange interaction \(\tau _{1}\) and spin-singlet interaction \(\tau _{2}\) , two conserved quantities, the number of atoms N and the total magnetization M. We first give a complete classification of ground state solutions and show the validity of single-mode approximation (SMA) phenomenon in \({\mathbb {R}}^{d}\) . In the one dimensional case, the energy functional is bounded from below on the related physical manifold, the ground states exist and are obtained as global minimizers. When \(d=2\) , the energy functional is not always bounded on the related physical manifold. We give a complete classification of the existence and nonexistence of global minimizers, and the explicit thresholds of existence and nonexistence of ground state solution were obtained. In the three dimensional case, the energy functional is always unbounded on the related physical manifold, 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 spin-2 BEC under the effect among spin-independent interaction, spin-exchange interaction and spin-singlet interaction but also support some experimental observations as well as numerical results on spin-2 BEC. Our results are the first studies on quantitative properties of ground states for spin-2 BEC.
