  Repulsive Bose-Einstein condensates immersed into a double-well trap
potential are studied within the framework of the recently introduced
mean-field approach which allows for bosons to reside in several different
orthonormal orbitals. In the case of a one-orbital mean-field theory
(Gross-Pitaevskii) the ground state of the system reveals a bifurcation
scenario at some critical values of the interparticle interaction and/or the
number of particles. At about the same values of the parameters the two-orbital
mean-field predicts that the system becomes two-fold fragmented. By applying
the three-orbital mean field we verify numerically that for the double-well
external potential studied here the overall best mean-field is achieved with
two orbitals. The variational principle minimizes the energy at a vanishing
population of the third orbital. To discuss the energies needed to remove a
boson from and the energies gained by adding a boson to the condensate, we
introduce boson ionization potentials and boson affinities and relate them to
the chemical potentials. The impact of the finite number of bosons in the
condensate on these quantities is analyzed. We recall that within the framework
of the multi-orbital mean-field theory each fragment is characterized by its
own chemical potential. Finally, the stability of fragmented states is
discussed in terms of the boson transfer energy which is the energy needed to
transfer a boson from one fragment to another.
