Comparaison between Coulomb and Hulth`en potentials within Bohr Hamiltonian for $gamma$-rigid nuclei in the presence of minimal length

In this work we solve the Schrodinger equation for Bohr Hamiltonian with Coulomb and Hulthen potentials within the formalism of minimal length in order to obtain analytical expressions for the energy eigenvalues and eigenfunctions by means of asymptotic iteration method. The obtained formulas of the energy spectrum and wave functions, are used to calculate excitation energies and transition rates of $gamma$-rigid nuclei and compared with the experimental data at the shape phase critical point X(3) in nuclei.