On the asymptotic methods for nuclear collective models

Contractions of orthogonal groups to Euclidean groups are applied to analytic descriptions of nuclear quantum phase transitions. The semiclassical asymptotic of multipole collective Hamiltonians are also investigated.

A solvable model which has X(5) as a limiting symmetry and removes some inherent drawbacks

Solvable Hamiltonians for the $beta$ and $gamma$ intrinsic shape coordinates are proposed. The eigenfunctions of the $gamma$ Hamiltonian are spheroidal periodic functions, while the Hamiltonian for the $beta$ degree of freedom involves the Davidsons potential and admits eigenfunctions which can be expressed in terms of the generalized Legendre polynomials. The proposed model goes to X(5) in the limit of $|gamma|$-small. Some drawbacks of the X(5) model, as are the eigenfunction periodicity and the $gamma$ Hamiltonian hermiticity, are absent in the present approach. Results of numerical applications to $^{150}$Nd, $^{154}$Gd and $^{192}$Os are in good agreement to the experimental data. Comparison with X(5) calculations suggests that the present approach provides a quantitative better description of the data. This is especially true for the excitation energies in the gamma band.