  Exactly solvable models provide a unique method, via qualitative changes in
the distribution of the ground-state roots of the Bethe Ansatz equations, to
identify quantum phase transitions. Here we expand on this approach, in a
quantitative manner, for two models of Bose--Einstein condensates. The first
model deals with the interconversion of bosonic atoms and molecules. The second
is the two-site Bose--Hubbard model, widely used to describe tunneling
phenomena in Bose--Einstein condensates. For these systems we calculate the
ground-state root density. This facilitates the determination of analytic forms
for the ground-state energy, and associated correlation functions through the
Hellmann--Feynman theorem. These calculations provide a clear identification of
the quantum phase transition in each model. For the first model we obtain an
expression for the molecular fraction expectation value. For the two-site
Bose--Hubbard model we find that there is a simple characterisation of
condensate fragmentation.
