Quasi-exact description of the γ-unstable shape phase transition

The equation of the gamma-unstable Bohr Hamiltonian, with particular forms of the sextic potential in the beta shape variable, is exactly solved for a finite number of states. The shape of the quasi-exactly solvable potential is then defined by the number of exactly determined states. The effect of exact solvability order on the spectral characteristics of the model is closely investigated, especially, concerning the critical point of the phase transition from spherical to deformed shapes. The energy spectra and the B(E2) transition probabilities, up to a scaling factor, depend only on a single-free parameter, while for the critical point, parameter-free results are available. Several numerical applications are done for nuclei undergoing a gamma-unstable shape phase transition in order to identify critical nuclei based on the most suitable exact solvability order.