  In this work, we investigate how both rotation and a Coulomb potential affect
the quantum mechanical description of a spin-$1/2$ particle in the presence of
the Aharonov-Bohm effect. We employ the method of the self-adjoint extensions
in the framework of the Pauli-Schr\"odinger equation. We discuss the role of
the spin degree of freedom on this problem, find the energy spectrum, and
investigate the results in detail.
