-$_{4}$ -$_{2}$ 2 4 x 0.2 0.4 0.6 0.8 1.0 $_{V}$H$_{x}$ L -$_{4}$ -$_{2}$ 2 4 x 0.2 0.4 0.6 0.8 1.0 $_{m}$H$_{x}$ L FIG. 1: Left, plot of the Woods Saxon potential ( W = 1). Right, plot of the position dependent mass function m$_{0}$ = 1. The parameters are: L = 2 and a = 10 (solid line), a = 5 (dashed line) and a = 3 (dot-dashed line). In the limit aL ≫ 1 the Woods-Saxon potential reduces to a smooth barrier approaching a square barrier. where γ = m$_{0}$/W . Therefore the position dependent mass problem could be looked at as that of a particle of constant mass m$_{0}$ coupling to both a vector potential ( V ( x )) and to a scalar potential S ( x ) = − γV ( x ). When discussing the barrier ( W > 0) it turns out that γ > 0 and therefore since S ( x ) + V ( x ) = (1 − γ ) V ( x ) there might be regions of the parameter space, m$_{0}$ = W (or m$_{0}$ ≈ W ) where the system is endowed with either an exact (or approximate) pseudo-spin symmetry which is defined by the condition S ( x ) + V ( x ) = constant . Such symmetry, the near equality of an attractive scalar potential with a repulsive vector potential is well know in the literature [27, 28] of the Dirac equation and has been proved very useful in describing the motion of nucleons in the relativistic mean fields resulting form nucleon-meson interactions, nucleon-nucleon Skyrme-type interactions and QCD sum rules. Further investigation of the possible consequences of this symmetry for the system under con- sideration goes however beyond the scope of the present work. D. Asymptotic expressions and boundary conditions of the scattering problem In the negative region from Eq. (19) we have φ$_{L}$ ( y ) = D$_{1}$ y $^{µ}$(1 − y ) − λ $_{2}$F$_{1}$ ( µ − ν − λ, µ + ν − λ ; 2 µ ; y ) + D$_{2}$ y 1 − $^{µ}$(1 − y ) − λ $_{2}$F$_{1}$ (1 − µ − ν − λ, 1 − µ + ν − λ ; 2 − 2 µ ; y ) . (26) 10