  We compute the Casimir energy for a system consisting of a fermion and a
pseudoscalar field in the form of a prescribed kink. This model is not exactly
solvable and we use the phase shift method to compute the Casimir energy. We
use the relaxation method to find the bound states and the Runge-Kutta-Fehlberg
method to obtain the scattering wavefunctions of the fermion in the whole
interval of $x$. The resulting phase shifts are consistent with the weak and
strong forms of the Levinson theorem. Then, we compute and plot the Casimir
energy as a function of the parameters of the pseudoscalar field, i.e. the
slope of $\phi(x)$ at x=0 ($\mu$) and the value of $\phi(x)$ at infinity
($\theta_0$). In the graph of the Casimir energy as a function of $\mu$ there
is a sharp maximum occurring when the fermion bound state energy crosses the
line of E=0. Furthermore, this graph shows that the Casimir energy goes to zero
for $\mu\rightarrow 0$, and also for $\mu\rightarrow \infty$ when $\theta_0$ is
an integer multiple of $\pi$. Moreover, the graph of the Casimir energy as a
function of $\theta_0$ shows that this energy is on the average an increasing
function of $\theta_0$ and has a cusp whenever there is a zero fermionic mode.
We finally compute the total energy of a system consisting of a valence fermion
in the ground state. Most importantly, we show that this energy (the sum of the
Casimir energy and the energy of the fermion) is minimum when the background
field has winding number one, independent of the details of the background
profile. Throughout the paper we compare our results with those of a simple
exactly solvable model, where a piece-wise linear profile approximates the
kink. We find that the kink is an almost reflectionless barrier for the
fermions, within the context of our model.
