  We study the ground-state properties of an atomic-molecular boson conversion
model through an exact Bethe Ansatz solution. For a certain range of parameter
choices, we prove that the ground-state Bethe roots lie on the positive
real-axis. We then use a continuum limit approach to obtain a singular integral
equation characterising the distribution of these Bethe roots. Solving this
equation leads to an analytic expression for the ground-state energy. The form
of the expression is consistent with the existence of a line of quantum phase
transitions, which has been identified in earlier studies. This line demarcates
a molecular phase from a mixed phase. Certain correlation functions, which
characterise these phases, are then obtained through the Hellmann-Feynman
theorem.
