Non-relativistic quantum effects of the topology of the spacetime with the distortion of a vertical line into a vertical spiral on the harmonic oscillator are investigated. By searching for analytical solutions to the Schrödinger equation in this topological defect background, it is shown that the topology of the spacetime modifies the spectrum of energy of the harmonic oscillator. Besides, it is shown that there exists an Aharonov-Bohm-type effect for bound states.

It is analysed Aharonov-Bohm-type effects when a spinless quantum particle is in an elastic medium with the distortion of a vertical line into a vertical spiral. By confining the spinless particle to a cylindrical box, the analogue of the Aharonov-Bohm effect for bound states is observed due to influence of the topological defect on the allowed energies of the system. It corresponds to the shift in the angular momentum quantum number yielded by the effects of the topology of the distortion of a vertical line into a vertical spiral. In addition, it is analysed the effects of rotation. It is shown that the Aharonov-Bohm effect for bound states exists. Besides, there is an analogue of the coupling between the angular momentum and angular velocity of the rotating frame.

We consider a spacetime with a distortion of a vertical line into a vertical spiral, thus, we analyse non-relativistic effects on the interaction of a point charge with a uniform magnetic field in this spacetime background. In addition, we consider the topological defect spacetime with an internal flux. Then, we analyse the Aharonov-Bohm effect for bound states in the non-relativistic limit by confining the point charge to a hard-wall confining potential, two cylindrical surfaces with rigid walls and a cylindrical surface.

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