\left[\left(\hat{p}_{0}^{\prime} - \frac{e E^{\prime}}{c} x^{\prime}_{1} \right)^{2} - \hat{p}^{\prime 2}_{1} - \hat{p}^{\prime 2}_{2} - \hat{p}^{\prime 3}_{2} \mp  i\frac{e \hbar  E^{\prime}}{c} \sigma_{1} \right] \phi^{\pm \prime }_{P} = 0 \, ,$$ and the basis set for the eigenspinors is given by the eigenvectors of $\sigma_{1}$. While the general structure of the differential equations is similar to the previous case, there are key differences. Notice first that the energy has become a continuous parameter that displaces the origin of $x_{1}^{\prime}$. The eigenfunctions are still given by a product of free particle states for the $x_{2}^{\prime}$ and $x_{3}^{\prime}$ components, but the $x_{1}^{\prime}$ component satisfies Weber’s differential equation and, as such, the corresponding eigenfunctions are given by parabolic cylinder functions $D_{a}$ [@Abramowitz_1972]. The eigenspinors are $$\begin{aligned}
\phi_{P1}^{\pm \prime} &= e^{i k_{\perp}^{\prime} x_{\perp}^{\prime}} D_{-a-1}\left( \xi^{\prime} \right)\left( \begin{array}{c}
1 \\
\pm 1 \\
\end{array} \right) \, , \label{eige_16a} \\
\phi_{P2}^{\pm \prime} &= e^{i k_{\perp}^{\prime} x_{\perp}^{\prime}} D_{-a}\left( \xi^{\prime} \right)\left( \begin{array}{c}
1 \\
\mp 1 \\
\end{array} \right) \, , \label{eige_16b}\end{aligned}$$ where $$\begin{aligned}
l_{\mathcal{E}}^{\prime} & = \sqrt{\frac{\hbar c}{  e E^{\prime}}} \, , \\
\xi' &=  \sqrt{2} e^{i \pi /4} \left( \upvarepsilon' l_{\mathcal{E}}^{\prime}/\hbar c - x_{1}^{\prime}/l_{\mathcal{E}}^{\prime} \right)\, , \label{eige_17}\\
a' &= -i l_{\mathcal{E}}^{\prime 2} \left( k_{2}^{\prime 2} + k_{3}^{\prime 2} \right) 