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Abstract.The equation of the Bohr-Mottelson Hamiltonian with a sextic oscillator potential is solved for γ-rigid prolate nuclei. The associated shape phase space is reduced to three variables which are exactly separated. The angular equation has the spherical harmonic functions as solutions, while the β equation is brought to the quasi-exactly solvable case of the sextic oscillator potential with a centrifugal barrier. The energies and the corresponding wave functions are given in closed form and depend, up to a scaling factor, on a single parameter. The 0 + and 2 + states are exactly determined, having an important role in the assignment of some ambiguous states for the experimental β bands. Due to the special properties of the sextic potential, the model can simulate, by varying the free parameter, a shape phase transition from a harmonic to an anharmonic prolate β-soft rotor crossing through a critical point. Numerical applications are performed for

An analytical solution for the Davydov-Chaban Hamiltonian with a sextic oscillator potential for the variable β and γ fixed to 30 • , is proposed. The model is conventionally called Z(4)-Sextic.For the considered potential shapes the solution is exact for the ground and β bands, while for the γ band an approximation is adopted. Due to the scaling property of the problem the energy and B(E2) transition ratios depend on a single parameter apart from an integer number which limits the number of allowed states. For certain constraints imposed on the free parameter, which lead to simpler special potentials, the energy and B(E2) transition ratios are parameter independent. The energy spectra of the ground and first β and γ bands as well as the corresponding B(E2) transitions, determined with Z(4)-Sextic, are studied as function of the free parameter and presented in detail for the special cases. Numerical applications are done for the 128,130,132 Xe and 192,194,196 Pt isotopes, revealing a qualitative agreement with experiment and a phase transition in Xe isotopes.

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The eigenvalue equation associated to the Bohr-Mottelson Hamiltonian is considered in the intrinsic reference frame and amended by replacing the harmonic oscillator potential in the β variable with a sextic oscillator potential with centrifugal barrier plus a periodic potential for the γ

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A diagonalization procedure for an SO(5) invariant Bohr Hamiltonian with a sextic potential for the β shape variable, having simultaneous spherical and deformed minima, is presented. The double-well potential allows for the study of shape coexistence and mixing phenomena from a geometrical perspective. The spectral and dynamical features of the model are investigated for different shapes of the potential amended with the centrifugal contribution from the kinetic term and subjected to the condition of degenerate minima. A generalization of the model is applied for the description of the low-lying energy spectra and shape mixing evolution in Mo nuclei. For each isotope, a shape transition from small to large deformation is found to occur between distinct states with various degrees of intensity and shape mixing.

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