  In this work, we derive a closed solution of the Shr$ \ddot{o} $dinger
equation for Bohr Hamiltonien within the minimal length formalism. This
formalism is inspired by Heisenberg algebra and a generlized uncertainty
principle (GUP), applied to the geometrical collective Bohr- Mottelson model
(BMM) of nuclei by means of deformed canonical commutation relation and the
Pauli-Podolsky prescription. The problem is solved by means conjointly of
asymptotic iteration method (AIM) and a quantum perturbation method (QPM) for
transitional nuclei near the critical point symmetry Z(4) corresponding to
phase transition from prolate to $\gamma$-rigid triaxial shape. A scaled
Davidson potentiel is used as a restoring potential in order to get physical
minimum. The agreement between the obtained theoretical results and the
experimental data is very satisfactory.
