We investigate the vacuum polarization and the Casimir energy of a Dirac field coupled to a scalar potential in one spatial dimension. Both of these effects have a common cause which is the distortion of the spectrum due to the coupling with the background field. Choosing the potential to be a symmetrical square-well, the problem becomes exactly solvable and we can find the whole spectrum of the system, analytically. We show that the total number of states and the total density remain unchanged as compared with the free case, as one expects. Furthermore, since the positive- and negative-energy eigenstates of the fermion are fermion-number conjugates of each other and there is no zero-energy bound state, the total density and the total number of negative and positive states remain unchanged, separately. Therefore, the vacuum polarization in this model is zero for any choice of the parameters of the potential. It is important to note that although the vacuum polarization is zero due to the symmetries of the model, the Casimir energy of the system is not zero in general. In the graph of the Casimir energy as a function of the depth of the well there is a maximum approximately when the bound energy levels change direction and move back towards their continuum of origin. The Casimir energy for a fixed value of the depth is a linear function of the width and is always positive. Moreover, the Casimir energy density (the energy density of all the negative-energy states) and the energy density of all the positive-energy states are exactly the mirror images of each other. Finally, computing the total energy of a valence fermion present in the lowest fermionic bound state, taking into account the Casimir energy, we find that the lowest bound state is almost always unstable for the scalar potential.
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