Energy Levels of a Bose-Einstein System of Particles with Attractive Interactions

An $N$-body Bose-Einstein system of particles with long-range attraction and hard-sphere repulsion between particles is considered. It is shown that if the constants of the interaction have values within a certain range it is possible to calculate the ground-state energy of the system as a function of $\frac{\Omega }{N}$, where $\Omega$ is the volume of the box containing the system, in the limit $N\to \infty$, $\Omega \to \infty$, with $\frac{N}{\Omega }=\rho$ fixed. The results show that the system can possess an $N$-body bound state, which has an equilibrium density and negative energy, and that the interactions can be saturating. Excited states are also considered. It is shown that low-lying excitations consist purely of phonons, whose velocity agrees with that computed from the macroscopic compressibility, furnished by the ground-state energy. The formula for the general excited energy levels suggests that thermodynamically the system may have a "gas" phase and two "liquid" phases, the transition between the two "liquid" phases being the analog of the Bose-Einstein condensation of the ideal gas. Thermodynamic considerations are, however, not contained in this paper.