Do Attractive Bosons Condense

Motivated by experiments on Bose atoms in traps which have attractive interactions (e.g., ${}^{7}\mathrm{Li}$), we consider two models which may be solved exactly. We construct the ground states subject to the constraint that the system is rotating with angular momentum proportional to the number of atoms. In a conventional system this would lead to quantized vortices; here, for attractive interactions, we find that the angular momentum is absorbed by the center of mass motion. Moreover, the state is uncondensed and is an example of a ``fragmented'' condensate discussed by Nozi\`eres and Saint James. The same models with repulsive interactions are fully condensed in the thermodynamic limit.