Fermions in the background of mixed vector-scalar-pseudoscalar square potentials

The general Dirac equation in 1+1 dimensions with a potential with a completely general Lorentz structure is studied. Considering mixed vector-scalar-pseudoscalar square potentials, the states of relativistic fermions are investigated. This relativistic problem can be mapped into a effective Schr\"{o}dinger equation for a square potential with repulsive and attractive delta-functions situated at the borders. An oscillatory transmission coefficient is found and resonant state energies are obtained. In a special case, the same bound energy spectrum for spinless particles is obtained, confirming the predictions of literature. We showed that existence of bound-state solutions are conditioned by the intensity of the pseudoscalar potential, which posses a critical value.