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The quantum motions of the non-relativistic particles interact with potentialin the presence of the Aharonov-Bohm (AB) flux field in a topological defectgeometry associated to screw dislocation, which corresponds to a distortion ofa vertical line into a vertical spiral are investigated. We derived the radialSchr\"{o}dinger wave equation with anharmonic oscillator potential(superposition of harmonic oscillator plus inverse square potential) and obtainthe eigenvalue solution through the Heun equation. The expressions for theground state energy level and the radial wave function for the radial mode$n=1$ are analyzed. Then, we utilize this eigenvalue solution to molecularpotential models (pseudoharmonic and shifted pseudoharmonic potentials) andpresented the eigenvalue solutions. We study the quantum system with inversesquare potential in the presence of the quantum flux field and obtained theground state energy level and the radial wave function through the sameprocedure. We show that the eigenvalue solutions are influenced by thetopological defect of screw dislocation characterized by the parameter $\beta$and the quantum flux field $\Phi_{AB}$ which causes the shift in the energyspectrum and shows the gravitational analogue to the Aharonov-Bohm effect.

Non-Relativistic Particles Interact with Potential Under the effect of  Quantum flux in a space-time with a distortion of a vertical line into a  vertical spiral