In this paper, we present exact solutions of the Dirac equation in static curved spacetime using two distinct algebraic approaches. The first method employs su(1,1) algebra operators together with the tilting transformation, enabling the derivation of the energy spectrum and eigenfunctions for both the Hydrogen atom and the Dirac–Morse oscillator. The second approach, based on the Schrödinger factorization method, extends the analysis to three representative potentials: the hydrogen atom, the Dirac–Morse oscillator, and a linear radial potential. Although structurally different from those obtained in the first method, the resulting operators in this approach also close the su(1,1) algebra and, through representation theory, yield the corresponding energy spectra and eigenfunctions.
Salazar-Ramírez et al. (Fri,) studied this question.