  The Hamiltonian for a small number, N <= 11, of bosons in a rapidly rotating
harmonic trap, interacting via a short range (contact potential) or a long
range (Coulomb) interaction, is studied via an exact diagonalization in the
lowest Landau level. Our analysis shows that, for both low and high fractional
fillings, the bosons localize and form rotating boson molecules (RBMs)
consisting of concentric polygonal rings. Focusing on systems with the number
of trapped atoms sufficiently large to form multi-ring bosonic molecules, we
find that, as a function of the rotational frequency and regardless of the type
of repulsive interaction, the ground-state angular momenta grow in specific
steps that coincide with the number of localized bosons on each concentric
ring. Comparison of the conditional probability distributions (CPDs) for both
interactions suggests that the degree of crystalline correlations appears to
depend more on the fractional filling nu than on the range of the interaction.
The RBMs behave as nonrigid rotors, i.e., the concentric rings rotate
independently of each other. At filling fractions nu < 1/2, we observe well
developed crystallinity in the CPDs (two-point correlation functions). For
larger filling fractions nu > 1/2, observation of similar molecular patterns
requires consideration of even higher-order correlation functions.
