  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. 
Casimir Energy for a Coupled Fermion-Kink System and its stability