Hartree solutions for the self-Yukawian boson sphere

The ground state of an N-boson assembly interacting through two-body attractive Yukawa forces is analyzed in the Hartree approximation. The solution, which is not universal, can be adequately characterized by a unique parameter μ̃ proportional to μ$N^{\mathrm{-}1}$, where ${\mu }^{\mathrm{-}1}$ is the range of the force. It is found that for μ̃ larger than a critical value, there is no bound solution for the system. Our solutions are shown to rigorously fulfill the virial theorem.