Ground state property of Bose-Einstein gas for arbitrary power low interaction

We study Bose-Einstein gas for an arbitrary power low interaction $C_{\alpha}r^{-\alpha}$. This is done by the Hartree Fock Bogoliubov (HFB) approach at $T \le T_{c}$ and the mean field approach at $T>T_{c}$. Especially, we investigate the ground state property of Bose gas interacting through the Van der Waals $-C_{6}r^{-6}$ plus $C_{3}r^{-3}$ interactions. We show that the ground state under this interaction is stable if the ratio of coupling constants is larger than that of the critical curve. We find that the $C_{3}r^{-3}$ term plays an important role for the stability of the ground state when the density of atoms becomes sufficiently large at low temperature. Further, using the numerical values of $C_{3}$ and $C_{6}$, we confirm that the ground state of alkali atoms are stable.