Investigation of ground state structures and phase separation under confinement is of great interest in spinor Bose Einstein Condensates (BEC). In this paper we show that, in general, within the Thomas-Fermi (T-F) approximation, the phase separation scenario of stationary states can be obtained including all the mixed states on an equal footing for a spin-1 condensate for any confinement. Exact analytical expressions of energy density, being independent of local mass density for all allowed states enables this general analysis under T-F approximation. We study here in details a particular case of spherically symmetric harmonic confinement as an example and show a wide range of potential phase separation scenario for anti-ferromagnetic and ferromagnetic interactions.

The ground state of a spin-1 Bose-Einstein condensate is selected based on the most energetically stable stationary state. It is well known that for the homogeneous condensate linear and quadratic term plays an important role to lift the degeneracy among the stationary states, giving a rich phase diagram. In this article, we investigate the ground state in absence of linear and quadratic Zeeman terms under realistic trapping potential. The spin-dependent interaction strength plays a key role in favoring one of the stationary states to have the lowest energy and thus producing the ground state of the system. We notice that the Thomas-Fermi approximated results predict that for anti-ferromagnetic condensates the energy difference between the competing stationary states is really small, requiring further analysis considering the full profile of the condensate. Thus, for the purpose of further refining results, we introduce a variational method which provides the full number density profile of the condensate with very good accuracy even for small condensates in 3dimensional isotropic harmonic confinement as well as in effective 1-dimensional harmonic trapping. Then we compare all the relevant physical parameters with those of Thomas-Fermi results.

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We report some corrections in a few domain-formation diagrams arising due to a sign correction in the energy density of PM/APM states and an additional term in the energy density of MF1/MF2 states. The entire general method developed and presented for the analysis of domain formation scenarios under the Thomas-Fermi approximation remains unaffected because that is independent of energy density expressions.

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