  In a k-dimensional system of weakly interacting Bose atoms trapped by a
spherically symmetric and harmonic external potential, an exact expression is
obtained for the rotating ground states at a fixed angular momentum. The result
is valid for arbitrary interactions obeying minimal physical requirements.
Depending on the sign of a modified scattering length, it reduces to either a
collective rotation or a condensed vortex state, with no alternative. The
ground state can undergo a kind of quantum phase transition when the shape of
the interaction potential is smoothly varied.
