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Amplifying on a proposal by O'Dell et al. for the realization ofBose-Einstein condensates of neutral atoms with attractive $1/r$ interaction,we point out that the instance of self-trapping of the condensate, withoutexternal trap potential, is physically best understood by introducingappropriate "atomic" units. This reveals a remarkable scaling property: thephysics of the condensate depends only on the two parameters $N^2 a/a_u$ and$\gamma/N^2$, where $N$ is the particle number, $a$ the scattering length,$a_u$ the "Bohr" radius and $\gamma$ the trap frequency in atomic units. Wecalculate accurate numerical results for self-trapping wave functions andpotentials, for energies, sizes and peak densities, and compare with previousvariational results. As a novel feature we point out the existence of a secondsolution of the extended Gross-Pitaevskii equation for negative scatteringlengths, with and without trapping potential, which is born together with theground state in a tangent bifurcation. This indicates the existence of anunstable collectively excited state of the condensate for negative scatteringlengths.

Bose-Einstein condensates with attractive 1/r interaction: The case of self-trapping