## 4 Discussion
One possible explanation for the group velocity behaviors shown in Figures 7 and 8 is related to the width of the linear-like potential being smaller than that of the rectangular-like potential of positive intensity, i.e., \(a=1/10\) and \(b=1/2\), since, when the simulation of the Kronig-Penney model is done for parameters in which \(b>a\), a trend in group velocity observed in Figures 7 and 8 is found. Figure 9 displays the group velocities and effective masses for \(w_{0}=100\), \(a=1\), and \(b=1/3\) for a linear-like potential with two slopes: negative (Figure 9A,B) and positive (Figure 9C,D). This figure suggests that the relationship between the parameters \(a\) and \(b\) determines the behavior of the group velocity. When \(a<b\), it has the behavior shown in Figures 7 and 8, while, when \(b<a\), the behavior is presented as in Figure 9[33; 34; 35]. Here, it should be stated that the Airy functions are not responsible for the observed divergence in the results because the used values are neither too small nor too large to provoke an appreciable divergence in the Airy functions. The Airy functions cannot be completely ruled out because they are improper integrals and cannot be expressed in terms of elementary functions; in other words, they are special functions [25; 36]. The width of the rectangle also plays an important role in the results of the linear-like potential, mainly regarding whether it slopes positively or negatively, as mentioned in the previous paragraphs. The reason behind this is that if the width of the rectangle is much smaller than the width of the line and the height of the rectangle is constant, then a pseudo-Dirac-delta potential would be simulated. This behavior was visualized when the simulations of the Dirac-delta potential were made, as shown in Figure S1. However, there is a major difference: the width is not infinitely small, and the height is not infinitely high. Table 2 shows the comparison of the positive and negative group velocities between the linear potentials. It is evident that the maximum of the group rate of the negative  
Figure 9: Numerical simulation with the values of \(w_{0}=100\), \(a=1\), \(b=1/3\) for the group velocity (**A**,**B**) and the effective mass (**C**,**D**) for the linear-like potential. The data were obtained for 32 energy values. (**A**,**C**) show data obtained for positive-sloping linear-like potential and (**B**,**D**) show data obtained for negative-sloping linear-like potential. sion of the energy of the sloping linear-like potential decays faster than the maximum of the group rate of the positive-sloping linear-like potential. Table 3 shows the comparison of the effective mass for the positive and negative linear-like potentials. The same is evident for the discontinuity energy; the negatively sloped linear-like potential decays faster than the positively sloped linear potential. In the case of triangular-like potentials, their group velocities and effective masses cannot be simulated correctly due to the divergence shown in Figures S2 and S3. Moreover, for the triangular and inverted triangular-like potential, the divergence in the transcendental energy equation is clearly observed. This divergence is the product of the Airy functions, which are defined by improper integrals and simulated numerically [37]. Now, a generalization for the periodic potential with any shape can be proposed, i.e., the generalization that can be developed for the rectangular-like potential by considering arbitrary N potentials of the form \(V=V(x)\) coupled in an interval \([-b,b]\). It should be noted that, interestingly, it has the same forms given in Equations (4) and (5) as can be seen in Equation (23).