  Bose-Einstein condensates in a double-well potential contain the essential
ingredients to study many-body systems within a rich classical phase-space that
includes an unstable point and a separatrix. Employing a selfconsistent finite
difference method, we study some of their quantum properties and their
dependency on the strength of the boson-boson interaction. We observe a
deviation in the critical parameters associated with a behavior change in both
the energy distribution and the eigenstates of the system. We also examine the
trends of the nonclassicality via the Wigner function, the tunneling
transmission coefficient, and the nonorthogonality of eigenstates associated
with the nonlinearity aspects of the Gross-Pitaevskii equation.
