  The energy spectrum for an ultracold rotating Bose gas in a harmonic trap is
calculated exactly for small systems, allowing the atoms to occupy several
Landau levels. Two vortex-like states and two strongly correlated states (the
Pfaffian and Laughlin) are considered in detail. In particular, their critical
rotation frequencies and energy gaps are determined as a function of particle
number, interaction strength, and the number of Landau levels occupied (up to
three). For the vortex-like states, the Lowest Landau level (LLL) approximation
is justified only if the interaction strength decreases with the number of
particles; nevertheless, the constant of proportionality increases rapidly with
the angular momentum per particle. For the strongly correlated states, however,
the interaction strength can increase with particle number without violating
the LLL condition. The results suggest that in large systems, the Pfaffian and
Laughlin states might be stabilized at rotation frequencies below the
centrifugal limit for sufficiently large interaction strengths, with energy
gaps a significant fraction of the trap energy.
