Supersymmetric approach to exact solutions of $(1+1)$-dimensional time-independent Klein-Gordon equation : Application to a position-dependent mass and a $\mathcal{PT}$-symmetric vector potential

Abstract: Rigorous use of SUSYQM approach applied for Klein-Gordon equation with scalar and vector potentials is discussed. The method is applied to solve exactly, for bound states, two models with position-dependent masses and $\mathcal{PT}$-symmetric vector potentials, depending on some parameters. The necessary conditions on the parameters to get physical solutions are described. Some special cases are also derived by adjusting the parameters of the models.