V. DISCUSSION AND CONCLUSIONS We have solved the scattering problem for the one-dimensional Dirac equation with the WS potential in the position-dependent mass formalism. We have set some conditions on the equation in order to keep the structure of the hypergeometric equation which give a suitable mass function. These conditions provide a first order differential equation which can be solved exactly. With the physical requirement that the mass function at infinity goes to a constant mass m$_{0}$ , specifies completely the mass function. Once the mass function has been found, c.f. Eq. 25, we have followed the same technique employed in ref. [24] solving the equation in the negative and positive region separately and giving the solution in terms of the hypergeometric function $_{2}$F$_{1}$ . For the scattering problem ordinary boundary conditions at infinity are imposed and then the match at x = 0 allows to specify all unknowns up to a normalization constant. We note that our method of solving the Dirac equation for a particular case of effective position dependent mass function has the drawback of providing a mass function which does not interpolate smoothly with the constant mass case. In other words our mass function does not contain a parameter such that when set to zero reduces the mass function to the constant case ( m$_{0}$ ). Further studies in this direction should be pursued in order to overcome such difficulties. We have obtained analytical expressions for the transmission and reflection coefficients, and we explicitly verified that unitarity ( R + T = 1) is preserved in the PDM case. We have also studied the bound states, i.e. the discrete spectrum of the WS potential well with the effective position dependent mass, finding an exact analytical condition for the energy eigenvalues (in the form of a transcendental equation which needs to solved numerically). We have provided an explicit numerical example finding the eigenvalues and the wave function for a specific choice of the parameters. Our approach offers one of the few examples where the Dirac equation is solved exactly in the position dependent mass case and in an external potential. To the best of our knowledge the only other example is reported in ref. [18] where the three-dimensional Dirac equation is solved for the PDM case in the Coulomb external field. We note a similarity between the present work and that reported in [18]. In both cases the mass function for which an exact solution is found shares similarities with the external potential. In [18] the spherically symmetric mass function for which the problem is solved is m ( r ) = 1 + µλ $^{2}$/r where, in atomic units, m$_{0}$ = ¯ h = 1 and λ is the Compton wavelength. Our mass function, c.f. Eq. 25, is also certainly related to the shape of the Woods-Saxon external potential. 22