## 4 Discussion  
From the periodic potentials here proposed, it has been possible to corroborate the hypothesis that they can approach the Kronig-Penney model, i.e., the rectangular-like potential with a negative intensity and linear-like potentials can also yield results similar to those found in the Kronig-Penney model. For triangular-like potentials, a divergence in numerical calculation occurred, avoiding additional testing, as shown in Figures S2 and S3. In Figure 5, it can be seen that the orientation of the group velocity in the first band energy changes for certain values of \(w\) (potential height), i.e., either for values less than or greater than \(w_{0}=100\). This may be explained considering that the rectangle at the interval \([-b,b]\) is of negative intensity. However, its energy dispersion is close to that found for the Kronig-Penney model, as can be seen in Figure 2.  
The energy \(H_{0}(E)\) function in Figure S4 expresses the variation in the group velocity in the rectangular-like potential of negative intensity, in agreement with the results shown in Figure 6. Therefore, the results in Figure S4 demonstrate that the orientation of the group velocity depends on which value of \(\cos\left(kT\right)=\pm 1\) is crossed first. Thus, if the rate of change is positive, then the group velocity is positive, and vice versa [33]. In addition, two general issues can be stated: (i) the change in the sign of the group velocity has been evidenced in the numerical simulation of the second energy band when performed at potentials \(V=V(x)\) that are close to a Dirac-delta potential, and (ii) the change in the orientation of the group velocity is not a particular case. Table 1 shows the orientation changes in the group velocity when varying each of the parameters of the periodic rectangular-like potential of negative intensity. The linear-like potentials also approximate the Kronig-Penney potential, as shown in Figures 3 and 4. However, on the right-hand side of these figures, it is observed that the potential rapidly approaches the one with a negative slope. For the group velocity using linear-like potentials, a peculiarity occurs in comparison with the results of the rectangular-like potential with negative intensity. For linear-like potentials, the numerically calculated group velocities for low and high values of \(w\) first show an increase in their group velocity until it reaches a maximum, whereby it subsequently decays for higher values of \(k\), as \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**First Energy Band** & \(w_{0}\) & \(\bullet\) & \(w\) & \(b\) \\ \hline \multirow{8}{*}{\(v_{G}\) positive} & 100 & 1/10 & 125 & 1/2 \\  & 100 & 1/10 & 25 & 1/2 \\  & 75 & 1/10 & 5 & 1/3 \\  & 75 & 1/10 & 5/2 & 1/3 \\  & 50 & 1/5 & 25 & 1/2 \\  & 50 & 1/5 & 150 & 1/2 \\  & 150 & 1/10 & 30 & 1/2 \\  & 150 & 1/10 & 150 & 1/2 \\  & 150 & 1/5 & 100 & 1/3 \\  & 150 & 1/5 & 120 & 1/3 \\ \hline \multirow{8}{*}{\(V_{G}\) negative} & 100 & 1/10 & 100 & 1/2 \\  & 100 & 1/10 & 10 & 1/2 \\  & 75 & 1/10 & 10 & 1/3 \\ \cline{1-1}  & 75 & 1/10 & 25 & 1/3 \\ \cline{1-1}  & 50 & 1/5 & 10 & 1/2 \\ \cline{1-1}  & 150 & 1/10 & 20 & 1/2 \\ \cline{1-1}  & 150 & 1/10 & 100 & 1/2 \\ \cline{1-1}  & 150 & 1/5 & 20 & 1/3 \\ \cline{1-1}  & 150 & 1/5 & 30 & 1/3 \\ \hline \hline \end{tabular}  
* Cases in which the group velocity changes its orientation in the first energy band for the rectangular-like potential of negative intensity.  
\end{table} 
Table 1: Group velocity for rectangular-like potential of negative intensity. shown in Figures 7 and 8. In other words, the group velocity curves show an asymmetric parabolic-like behavior, with their maxima slightly depending on the \(w\)-value.