  The behaviour of massive fermions is analyzed with scalar and vector
potentials. A continuous chiral-conjugation transformation decouples the
equation for the upper component of the Dirac spinor provided the vector
coupling does not exceed the scalar coupling. It is shown that a
Sturm-Liouville perspective is convenient for studying scattering as well as
bound states. One possible isolated solution (excluded from the Sturm-Liouville
problem) corresponding to a bound state might also come into sight. For
potentials with kink-like profiles, beyond the intrinsically relativistic
isolated bound-state solution corresponding to the zero-mode solution of the
massive Jackiw-Rebbi model in the case of no vector coupling, a finite set of
bound-state solutions might appear as poles of the transmission amplitude in a
strong coupling regime. It is also shown that the possible isolated bound
solution disappears asymptotically as the magnitude of the scalar and vector
coupling becomes the same. Furthermore, we show that due to the sizeable mass
gain from the scalar background the high localization of the fermion in an
extreme relativistic regime is conformable to comply with the Heisenberg
uncertainty principle.
