Exact Solutions of Klein--Gordon Equation with Scalar and Vector Rosen--Morse-Type Potentials

We obtain an exact analytical solution of the Klein--Gordon equation for the equal vector and scalar Rosen--Morse and Eckart potentials as well as the parity-time (PT) symmetric version of the these potentials by using the asymptotic iteration method. Although these PT symmetric potentials are non-Hermitian, the corresponding eigenvalues are real as a result of the PT symmetry.