Bose-Einstein atoms in atomic traps with predominantly attractive two-body interactions

Using the Perron-Frobenius theorem, we prove that the results by Wilkin, Gunn, and Smith [Phys. Rev. Lett. 80, 2265, (1998)] for the ground states at angular momentum L of N harmonically trapped Bose atoms, interacting via weak attractive δ²(r) forces, are valid for a broad class of predominantly attractive interactions V(r), not necessarily attractive for any r. This class is described by sufficient conditions on the two-body matrix elements of the potential V(r). It includes, in particular, the Gaussian attraction of arbitrary radius, -1/r-Coulomb and log(r)-Coulomb forces, as well as all the short-range interactions satisfying inequality ∫d²r⃗V(r)<0. In the precollapse regime, the angular momentum L is concentrated in the collective “center-of-mass” mode, and there is no condensation at high L.