Exact solutions of the Dirac oscillator under the influence of the Aharonov–Casher effect in the cosmic string background

In this paper, we study the influence of the Aharonov–Casher effect and of an external electromagnetic field on the relativistic and nonrelativistic energy spectra of the (3+1) -dimensional Dirac oscillator (DO) with magnetic dipole moment in two different scenarios: the Minkowski spacetime (flat case) and the cosmic string spacetime (curved case). Solving the DO, we first determine the flat relativistic spectrum, where we verified that such a spectrum depends explicitly on the magnetic energy E_m , quantum numbers n and l , quantum phase Φ _AC , and of an effective frequency Ω , on which it depends on the frequency of the DO, given by ω , and of a type of cyclotron frequency, given by ω _AC . So, due to the presence of the quantum phase, we note that the spectrum is a periodic function with periodicity ± 2 π . Besides, the function of E_m is to decrease or increase the energies depending on the choice of spin, while the function of ω _AC is to increase the energies of the particle and decrease those of the antiparticle. Solving the DO again, but now in the cosmic string spacetime, we determine the curved relativistic spectrum, where some terms have been modified due to the deficit angle η (curvature or topological parameter) of the cosmic string. We note that this deficit angle has the function of breaking the degeneracy of energy levels as well as increasing the values of such levels. Finally, we also analyzed the nonrelativistic limit of our results, and comparing our problem with other works, we verified that our results generalize some particular cases in the literature.