First branch of liquid states of a many-atom Bose system

We present more detailed properties of the first branch of the liquid states as suggested and studied recently, including the equilibrium density and the equilibrium energy-per-particle of the liquid, speed of phonons, and pair correlation functions. Results are presented both for liquid branches corresponding to negative scattering lengths\footnote{B. Gao, J. Phys. B \textbf{37}, L227 (2004).} and for those corresponding to positive scattering lengths\footnote{B. Gao, Phys. Rev. Lett. \textbf{95}, 240403 (2005).}.