Exact solution of the Dirac equation for a Coulomb and scalar potentials in the gravitational field of a cosmic string

We consider the problem of a relativistic electron in the presence of a Coulomb and scalar potentials in the background spacetime generated by a cosmic string. The scalar potential arises from the self-interaction potential which is due to the conical geometry of the spacetime of a cosmic string. We find the solution of the corresponding Dirac equation and we determine the energy spectrum of the particle. Some connection with a hydrogen atom in the spacetime generated by a cosmic string is discussed.