u (3)-Boson Model of Nuclear Collective Motion

The u(3)-boson model encompasses both the Elliott su(3) and Bohr-Mottelson collective models. The microscopic foundation of the u(3)-boson model is achieved by showing that it is the liquid (i.e., many nucleon) limit of the microscopic collective sp(3,$R$) theory. For ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ and ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}$ we construct microscopic vibrational states in the sp(3,$R$) theory which compare remarkably well with the simpler u(3)-boson prediction.