The ground and resonant states of three self-gravitating bosons and fermions

Recently, a new property of the Bose-Einstein condensates (BECs) has been proposed [1] that it is self bound for sufficiently strong self-gravitation, opening up a door to study self-gravitating systems with attractive $1/r$ potentials. Such an attractive potential can simulate gravity between the quantum systems. In the usual strong anisotropy regime, the inter-atomic potential takes a form of --u/r, where u is the coupling constant dependent on the laser intensity. Recently, we have investigated the ground state, excited states, and resonance states for three self-gravitating bosons and fermions using highly correlated exponential basis functions [2]. Our calculated bound states energies are lower than the earlier results in the literature [3]. We have also employed the complex-coordinate rotation method [4] to calculate the energies and widths for resonances lying below the $N=$2 and $N=$3 thresholds of the two-body subsystems [2]. [1] D. O'Dell\textit{ et al}., Phys. Rev. Lett. \textbf{84}, 5687 (2000); Phys. Rev. A \textbf{63}, 031603 (2001). [2] S. Kar and Y. K. Ho, Phys. Rev. A \textbf{76}, 032711(2007); Phys. Lett.$ A$\textbf{370}, 306 (2007) ; to be published. [3] J. P. D'Incao\textit{ et al.}, \textit{Phys. Rev. A} \textbf{75}, 032503 (2007). [4] Y. K. Ho, \textit{Phys. Reports} \textbf{99}, 1 (1983).