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The behaviour of fermions in the background of a double-step potential isanalyzed with a general mixing of scalar and vector couplings via continuouschiral-conjugation transformation. Provided the vector coupling does not exceedthe scalar coupling, a Sturm-Liouville approaching for the double-steppotential shows that the transmission coefficient exhibits oscillations andthat a finite set of intrinsically relativistic bound-state solutions mightappear as poles of the transmission amplitude in a strong coupling regime. Anisolated bound-state solution resulting from coupled first-order equationsmight also come into sight. It is also shown that all those possible boundsolutions disappear asymptotically as one approaches the conditions for therealization of the so-called spin and pseudospin symmetries in afour-dimensional space-time. Furthermore, we show that due to the additionalmass acquired by the fermion from the scalar background the high localizationof the fermion in an extreme relativistic regime does not violate theHeisenberg uncertainty principle.

Fermions in a mixed vector-scalar double-step potential via continuous chiral transformation