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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 

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