## 4 Discussion
\[\cos\left(kT\right)=\sinh\left(\beta a\right)\left[\frac{H_{1}^{2}(E)-\beta^{2 }H_{4}^{2}(E)}{2\beta H_{0}^{0}(E)}\right]+\cosh\left(\beta a\right)\left[ \frac{H_{2}^{2}(E)-H_{4}^{1}(E)}{2H_{0}^{0}(E)}\right] \tag{23}\]  
In turn, it can also be generalized for the zero-like potential coupled to arbitrary N potentials of the form \(V=V(x)\), as shown in Equation (24). \[\cos\left(kT\right)=\sin\left(\alpha a\right)\left[\frac{H_{1}^{2}(E)+\alpha^{ 2}H_{4}^{2}(E)}{2\alpha H_{0}^{0}(E)}\right]+\cos\left(\alpha a\right)\left[ \frac{H_{2}^{2}(E)-H_{4}^{1}(E)}{2H_{0}^{0}(E)}\right] \tag{24}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Group Velocity** & \(\mathbf{w}\) & \(\mathbf{kT}\) & \(\mathbf{v_{G}}\) \\ \hline \multirow{4}{*}{Positive slope linear potential} & 100 & 1.62 & 0.01550 \\  & 75 & 1.62 & 0.00835 \\  & 50 & 1.62 & 0.00456 \\  & 25 & 1.62 & 0.00257 \\  & 10 & 1.62 & 0.00192 \\ \hline \multirow{4}{*}{Negative slope linear potential} & 100 & 1.62 & 0.003110 \\  & 75 & 1.62 & 0.001920 \\ \cline{1-1}  & 50 & 1.62 & 0.001160 \\ \cline{1-1}  & 25 & 1.62 & 0.000701 \\ \cline{1-1}  & 10 & 1.62 & 0.000544 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Group velocity for linear potentials with two slopes: positive and negative. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Effective Mass** & \(\mathbf{w}\) & \(\mathbf{kT}\) & \(\mathbf{E}\) \\ \hline \multirow{4}{*}{Positive slope linear potential} & 100 & 1.57 & 62.2 \\  & 75 & 1.57 & 52.5 \\  & 50 & 1.57 & 41.9 \\  & 25 & 1.57 & 30.2 \\  & 10 & 1.58 & 22.1 \\ \hline \multirow{4}{*}{Negative slope linear potential} & 100 & 1.57 & 59.4 \\  & 75 & 1.57 & 49.9 \\ \cline{1-1}  & 50 & 1.57 & 39.3 \\ \cline{1-1}  & 25 & 1.57 & 27.6 \\ \cline{1-1}  & 10 & 1.57 & 19.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Effective mass for linear potentials with two slopes: positive and negative. An interesting point about Equations (23) and (24) is that they have a similar form to Equations (2) and (3) for particular cases and to the equations of the transcendental energy equation when approximating to the Dirac-delta potential [30]. The elements of the matrix \(H(E)\) shown in Equations (23) and (24) represent energy determinants, which are the result of [PERSON]'s theorem for N + 1 potentials, being a rectangular-like potential or, alternatively, a zero-like potential. The equation of the matrix \(H(E)\) is given by Equation (25). \[H(E)= \begin{bmatrix}H_{1}^{1}(E)&H_{2}^{2}(E)&H_{3}^{3}(E)\\ H_{2}^{1}(E)&H_{2}^{2}(E)&H_{3}^{3}(E)\\ H_{3}^{1}(E)&H_{3}^{2}(E)&H_{3}^{3}(E)\\ H_{4}^{1}(E)&H_{4}^{2}(E)&H_{4}^{3}(E)\end{bmatrix}  \tag{25}\]  
With the obtained results, a generalized theorem can be proposed for periodic potentials having the studied form; see Appendices A and B. This theorem is of great importance because it helps to simplify the mathematical calculations of arbitrary N-coupled potentials to a potential with known wave functions. Equations (26) and (27) provide this energy function. \[\Gamma(E) = -\Lambda(E) \tag{26}\] \[\Gamma(E) = \frac{H_{4}^{2}(E)}{2H_{0}^{0}(E)W\{y_{n+1},z_{n+1}\}}\Big{[}z_{n +1}^{{}^{\prime}}(b_{n+1})y_{n+1}^{{}^{\prime}}(b_{m})-z_{n+1}^{{}^{\prime}}( b_{m})y_{n+1}^{{}^{\prime}}(b_{n+1})\Big{]}\] (27) \[+ \frac{H_{1}^{2}(E)}{2H_{0}^{0}(E)W\{y_{n+1},z_{n+1}\}}\big{[}z_{n +1}(b_{n+1})y_{n+1}(b_{m})-z_{n+1}(b_{m})y_{n+1}(b_{n+1})\big{]}\] \[+ \frac{H_{2}^{2}(E)}{2H_{0}^{0}(E)W\{y_{n+1},z_{n+1}\}}\Big{[}-z_{ n+1}^{{}^{\prime}}(b_{n+1})y_{n+1}(b_{m})+z_{n+1}(b_{m})y_{n+1}^{{}^{\prime}}( b_{n+1})\Big{]}\] \[+ \frac{H_{4}^{1}(E)}{2H_{0}^{0}(E)W\{y_{n+1},z_{n+1}\}}\Big{[}-z_{ n+1}(b_{n+1})y_{n+1}^{{}^{\prime}}(b_{m})+z_{n+1}^{{}^{\prime}}(b_{m})y_{n+1}(b_{ n+1})\Big{]}\]  
where \(y_{n+1}(x)\) and \(z_{n+1}(x)\) in Equation (27) are the wave functions of the potential coupled to the arbitrary N potentials.