Quasi-exact solution of the Dirac equation on curved space-time with Coulomb scalar and vector potentials and Mie-type tensor potential with pseudo-spin and spin symmetries

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In this paper we solve the Dirac equation with the Coulomb scalar U(r) and vector V(r) potentials and type-Mie tensor potential in curved space-time whose metric is of type with spherical symmetry. For this we consider two types of symmetry in the system, first spin symmetry with V(r) = U(r) and then with pseudo-spin symmetry V(r) = − U(r). In both cases we have a tensor potential A(r) that coupling through the electromagnetic field A μ = (V(r), cA(r), 0, 0). We find, quasi-exactly, the Dirac spinor and the energy spectrum of the system, and we sketch the probability densities and some energy spectra.