a linear combination of Pauli spinors: $$\begin{aligned}
\phi_{D}^{+ \prime} & = e^{i  k_{\perp}' x_{\perp}'}  \left( \begin{array}{c}
i H_{\vert n \vert} \left( x_{1}'/l_{\mathcal{B}}'+ l_{\mathcal{B}}' k_{2}' \right)  \\
c^{\prime}_{\pm} H_{\vert n \vert +1 } \left( x_{1}'/l_{\mathcal{B}}'+ l_{\mathcal{B}}' k_{2}' \right) \\
\end{array} \right) \, , \label{eige_8a}\\
\phi_{D}^{- \prime} & = e^{i  k_{\perp}' x_{\perp}' }  \left( \begin{array}{c}
c^{\prime}_{\pm}  H_{\vert n \vert} \left( x_{1}'/l_{\mathcal{B}}'+ l_{\mathcal{B}}' k_{2}' \right)  \\
i H_{\vert n \vert +1 } \left( x_{1}'/l_{\mathcal{B}}'+ l_{\mathcal{B}}' k_{2}' \right) \\
\end{array} \right) \, , \label{eige_8b}\end{aligned}$$ with the relative weights depending on the energy and momentum through $$\label{eige_9}
c^{\prime}_{\pm} = \pm \sqrt{\frac{\upvarepsilon_{n,\pm}^{\prime} + \hbar c k_{3}^{\prime}}{\upvarepsilon_{n,\pm}^{\prime} - \hbar c k_{3}^{\prime}}} \, .$$ Finally, the solutions are transformed back to the laboratory frame, where the energies are found to be $$\label{eige_10}
\upvarepsilon_{n, \pm}=  \hbar c \left[ \beta_{\mathcal{B}} k_{2}  \pm \sqrt{\frac{\left(1 - \beta_{\mathcal{B}}^{2} \right)^{3/2}}{l^{2}_{\mathcal{B}}} \left(2 \vert n \vert + 2 \right)  +  k_{3}^{2}} \right] ,$$ while the eigenstates are \[eige\_11t\] $$\begin{aligned}
\phi_{D}^{+}  &= e^{i k_{\perp} x_{\perp}} \left( \begin{array}{c} i \Xi_{\mathcal{B}}^{+} H_{\vert n \vert} (z) - ic_{\pm} \Xi_{\mathcal{B}}^{-} H_{\vert n \vert +1} (z)  \\
c_{\pm} \Xi_{\mathcal{B}}^{+} H_{\vert n \vert +1 } (z) - \Xi_{\mathcal{B}}^{+} H_{\vert n \vert} (z)      \\
\end{array} \right) , \label{eige_11a}\\
\phi_{D}^{-}  &= e^{i k_{\perp} x_{\perp}} \left( 