  Motivated by the recent development of the Feshbach technique, we studied the
ground and low-lying excited states of attractive Bose-Einstein condensates on
a one-dimensional ring as a function of the strength of interactions. The
Gross-Pitaevskii mean-field theory predicts a quantum phase transition between
a uniform condensate and a bright soliton, and a gapless singular cusp in the
Bogoliubov excitation spectrum at the critical point. However, the exact
diagonalization reveals the presence of an energy gap at the critical point,
where the singularity is smeared by quantum fluctuations.
