  In a recent paper we studied an equation (called the "simple equation")
introduced by one of us in 1963 for an approximate correlation function
associated to the ground state of an interacting Bose gas. Solving the equation
yields a relation between the density $\rho$ of the gas and the energy per
particle. Our construction of solutions gave a well-defined function $\rho(e)$
for the density as a function of the energy $e$. We had conjectured that
$\rho(e)$ is a strictly monotone increasing function, so that it can be
inverted to yield the strictly monotone increasing function $e(\rho)$. We had
also conjectured that $\rho e(\rho)$ is convex as a function of $\rho$. We
prove both conjectures here for small densities, the context in which they have
the most physical relevance, and the monotonicity also for large densities.
Both conjectures are grounded in the underlying physics, and their proof
provides further mathematical evidence for the validity of the assumptions
underlying the derivation of the simple equation, at least for low or high
densities, if not intermediate densities, although the equation gives
surprisingly good predictions for all densities $\rho$. Another problem left
open in our previous paper was whether the simple equation could be used to
compute accurate predictions of observables other than the energy. Here, we
provide a recipe for computing predictions for any one- or two-particle
observables for the ground state of the Bose gas. We focus on the condensate
fraction and the momentum distribution, and show that they have the same low
density asymptotic behavior as that predicted for the Bose gas. Along with the
computation of the low density energy of the simple equation in our previous
paper, this shows that the simple equation reproduces the known and conjectured
properties of the Bose gas at low densities.
