In this paper, we analyze the relativistic quantum motion of a charged scalar particles in the presence of a Aharonov-Bohm and Coulomb potentials in the spacetimes produced by an idealized cosmic string and global monopole. We have calculated and discussed the eigensolutions of DKP equation and their dependence in both the geometry of the spacetimes and coupling constants parameters.

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We have analyzed the effects of a simple wormhole, known as the Ellis–Bronnikov-type wormhole, on a scalar field, where, analytically, we determine solutions of bound states and show that the relativistic energy profile of this scalar field is drastically influenced by the topology of space-time characterized by the presence of a global monopole. Before this analysis, we investigated the effects of this background on the deflection of light, which is influenced by the parameters associated with the wormhole throat and the topological defect.

Using a formulation of quantum mechanics based on orthogonal polynomials in the energy and physical parameters, we present a method that gives the class of potential functions for exactly solvable problems corresponding to a given energy spectrum. In this work, we study the class of problems associated with the continuous dual Hahn polynomial. These include the one-dimensional logarithmic potential and the three-dimensional Coulomb plus linear potential.

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