### Appendix A.2  
**Corollary A2 (Corollary of Rectangular Potential Generalization for One-Dimensional Periodic Potentials). **_This corollary expresses the particular case in which the potential coupled tothe arbitrary N potentials is rectangular. The equations corresponding to this approximation range from Equation (A10) to Equation (A15)._  
\[\cos\left(kT\right) =\sinh\left(\beta a\right)\left[\frac{H_{1}^{2}(E)-\beta^{2}H_{4}^{2 }(E)}{2\beta H_{0}^{0}(E)}\right]+\cosh\left(\beta a\right)\left[\frac{H_{2}^{2 }(E)-H_{4}^{1}(E)}{2H_{0}^{0}(E)}\right]\] (A10) \[f_{r}(E) =\frac{H_{1}^{2}(E)-\beta^{2}H_{4}^{2}(E)}{2\beta H_{0}^{0}(E)}\] (A11) \[g_{r}(E) =\frac{H_{2}^{2}(E)-H_{4}^{1}(E)}{2H_{0}^{0}(E)}\] (A12) \[\Gamma_{r}(E) =\sinh\left(\beta a\right)\left[\frac{H_{1}^{2}(E)-\beta^{2}H_{4} ^{2}(E)}{2\beta H_{0}^{0}(E)}\right]+\cosh\left(\beta a\right)\left[\frac{H_{2 }^{2}(E)-H_{4}^{1}(E)}{2H_{0}^{0}(E)}\right]\] (A13) \[\frac{d\Gamma_{r}(E)}{dE} =\left(\frac{df_{r}(E)}{dE}+ag_{r}(E)\frac{d\beta(E)}{dE}\right) \sinh\left(\beta(E)a\right)+\left(\frac{dg_{r}(E)}{dE}+af_{r}(E)\frac{d\beta( E)}{dE}\right)\cosh\left(\beta(E)a\right)\] (A14) \[\frac{d^{2}\Gamma_{r}(E)}{dE^{2}} =\left[ag_{r}(E)\frac{d^{2}\beta(E)}{dE^{2}}+2a\frac{dg_{r}(E)}{ dE}\frac{d\beta(E)}{dE}\right]\sinh\left(\beta(E)a\right)+\left[\frac{d^{2}f_{r}(E) }{dE^{2}}+a^{2}f_{r}(E)\left(\frac{d\beta(E)}{dE}\right)^{2}\right]\sinh\left( \beta(E)a\right)\] \[+\left[af_{r}(E)\frac{d^{2}\beta(E)}{dE^{2}}+2a\frac{df_{r}(E)}{ dE}\frac{d\beta(E)}{dE}\right]\cosh\left(\beta(E)a\right)+\left[\frac{d^{2}g_{r}(E) }{dE^{2}}+a^{2}g_{r}(E)\left(\frac{d\beta(E)}{dE}\right)^{2}\right]\cosh\left( \beta(E)a\right)\] (A15)  
**Proof of Corollary A2. ** The proof of this corollary is shown in Section S7 of the Supplementary Materials. \(\square\)  
The energy functions \(f_{r}(E)\) and \(g_{r}(E)\) are generalized energy functions, which simplify the calculation of the group velocity and effective mass at the generalized rectangular potential. *Corollary A3** (Corollary of Group Velocity and Effective Mass in One-Dimensional Periodic Potentials). : _This corollary states that the group velocity and effective mass of any periodic potential with a period T, composed of arbitrary N-coupled potentials to a potential with known wave functions, will then have an extreme value for the group velocity and a discontinuity point for the effective mass. These values can be computed using Equation (A16)._  
\[0=\frac{d^{2}\Gamma(E)}{dE^{2}}+\frac{1}{\sin\left(kT\right)\tan\left(kT \right)}\left(\frac{d\Gamma(E)}{dE}\right)^{2}\] (A16)  
**Proof of Corollary A3. ** The proof of this corollary is shown in Section S6 of the Supplementary Materials. \(\square\)  
The group velocity can reach a maximum or a minimum as a function of the energy band, as evidenced by the group velocity in Ref. [30]. Solving Equation (A16) provides the value of the wavenumber at which the extreme of the group velocity is obtained. Graphically speaking, the value of the wavenumber at which the discontinuity occurs is, interestingly, the same as that at which the extreme value of the group velocity is obtained. This is important because it allows the relation of the most important values of group velocity and effective mass to a single wavenumber value. *Corollary A4** (Corollary Approximation for the Dirac-Delta Potential in One-Dimensional Periodic Potentials). : _This corollary states that if, in Corollary 1, the N potentials have the peculiarity that the global minima are at the extremes and the global maximum is located at the midpoint, then the Dirac-delta periodic potential can be approximated by the following limits of Equations (A17) and (A18)._  
\[\lim_{(b,w)\rightarrow(0,\infty)}(\alpha f_{0}(E)) =\lim_{(b,w)\rightarrow(0,\infty)}\left(\frac{H_{1}^{2}(E)+\alpha ^{2}H_{4}^{2}(E)}{2H_{0}^{0}(E)}\right)=F_{d}(b,w) \tag{17}\] \[\lim_{(b,w)\rightarrow(0,\infty)}(g_{0}(E)) =\lim_{(b,w)\rightarrow(0,\infty)}\left(\frac{H_{2}^{2}(E)-H_{4} ^{1}(E)}{2H_{0}^{0}(E)}\right)=1 \tag{18}\]  
Proof of Corollary A4.