Analysis of a simple equation for the ground state of the Bose gas II: Monotonicity, Convexity and Condensate Fraction

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.