The spectral properties of a tractable collective model Hamiltonian are studied. The potential energy is truncated up to quartic terms in the quadrupole deformation variables, incorporating vibrational, $gamma$-independent rotational and axially defo
rmed rotational structures. These physically significant limits are analysed in detail and confronted with well-established approximation schemes. Furthermore, transitional Hamiltonians in between the limits are presented and discussed. All results are obtained within a recently presented Cartan-Weyl based framework to calculate $SU(1,1)times SO(5)$ embedded quadrupole collective observables.
In this article we report the results of detailed interacting boson model calculations with configuration mixing for the neutron-deficient Pb isotopes. Calculated energy levels and $B(E2)$ values for $^{188-196}$Pb are discussed and some care is sugg
ested concerning the current classification on the basis of level systematics of the $4_1^+$ and $6_1^+$ states in $^{190-194}$Pb. Furthermore, quadrupole deformations are extracted for $^{186-196}$Pb and the mixing between the different families (0p-0h, 2p-2h, and 4p-4h) is discussed in detail. Finally, the experimental and the theoretical level systematics are compared.
The case of U(5)--$hat{Q}(chi)cdothat{Q}(chi)$ mixing in the configuration-mixed Interacting Boson Model is studied in its mean-field approximation. Phase diagrams with analytical and numerical solutions are constructed and discussed. Indications for
first-order and second-order shape phase transitions can be obtained from binding energies and from critical exponents, respectively.
