  The ground and low-lying collective states of a rotating system of $N=3$
bosons harmonically confined in quasi-two-dimension and interacting via
repulsive finite-range Gaussian potential is studied in weakly to moderately
interacting regime. The $N$-body Hamiltonian matrix is diagonalized in
subspaces of quantized total angular momenta $0\le L \le 4N$ to obtain the
ground and low-lying eigenstates. Our numerical results show that breathing
modes with $N$-body eigenenergy spacing of $2\hbar\omega_{\perp}$, known to
exist in strictly 2D system with zero-range ($\delta$-function) interaction
potential, may as well exist in quasi-2D system with finite-range Gaussian
interaction potential. To gain an insight into the many-body states, the von
Neumann entropy is calculated as a measure of quantum correlation and the
conditional probability distribution is analyzed for the internal structure of
the eigenstates. In the rapidly rotating regime the ground state in angular
momentum subspaces $L=\frac{q}{2}N\left(N-1\right)$ with $q=2, 4$ is found to
exhibit the anticorrelation structure suggesting that it may variationally be
described by a Bose-Laughlin like state. We further observe that the first
breathing mode exhibits features similar to the Bose-Laughlin state in having
eigenenergy, von Neumann entropy and internal structure independent of
interaction for the three-boson system considered here. On the contrary, for
eigenstates lying between the Bose-Laughlin like ground state and the first
breathing mode, values of eigenenergy, von Neumann entropy and internal
structure are found to vary with interaction.
