  In this work, we investigate the quantum motions of non-relativistic
particles interacting with a potential in the presence of the Aharonov-Bohm
(AB) flux field within a topological defect geometry, for example, space-time
with a distortion of a vertical line into a vertical spiral. We begin by
deriving the radial Schr\"odinger wave equation, incorporating an anharmonic
oscillator potential, which is a superposition of a harmonic oscillator and an
inverse square potential, along with a constant term. The eigenvalue solution
is obtained through the confluent Heun equation focusing on the ground state
energy level and the radial wave function for the radial mode $n=1$ as an
example and analyze the results. Subsequently, we use these results in
molecular potential models, considering pseudoharmonic and shifted
pseudoharmonic potentials. The derived eigenvalue solutions provide insights
into the behavior of particles within these potentials. Expanding our
exploration, we study the quantum system featuring only an inverse square
potential in the presence of the quantum flux field in the same geometry
background. Employing the same procedure, we determine the ground state energy
level and the radial wave function. Notably, our findings reveal that the
eigenvalue solutions are significantly influenced by the topological defect
characterized by the parameter $\beta$, and the quantum flux field $\Phi_{AB}$.
This influence manifests as a shift in the energy spectrum, drawing parallels
to the gravitational analog of the Aharonov-Bohm effect.
