  Using the conformable fractional calculus, a new formulation of the Bohr
Hamiltonian is introduced. The conformable fractional energy spectra of free-
and two- parameters anharmonic oscillator potentials are investigated. The
energy eigenvalues and wave functions are calculated utilizing the
finite-difference discretization method. It is proved that the conformable
fractional spectra of the free-parameter Bonatsos potentials, $\frac{\beta ^{2
n}}{2}$, close completely the gaps between the classical spectra of the
vibrational U(5) dynamical symmetry, the $E(5)-\beta ^{2 n}$ models, and the
E(5) critical point symmetry. The ground effective sextic potential, which
generates both the ground state and the $\beta$ excited states $0^+$, is
considered to have two degenerate minima. In this case, the conformable
fractional spectra of sextic potentials show a change, as a function of barrier
height, from $\gamma$-unstable O(6) energy level sequence to the spectrum of
$E(5)-\beta ^{6}$ model and simultaneously provide new features. The shape
coexistence phenomena, in the ground band states, are identified. The energy
spectrum and shape coexistence with mixing phenomena in $\, ^{96}\text{MO}$
nucleus are discussed in the framework of the conformable fractional Bohr
Hamiltonian.
