/2 \, . \label{eige_18}\end{aligned}$$ The Weber differential equation is related to the harmonic oscillator equation by flipping the sign of the potential, a change that leads to imaginary values in the definition of the Pauli spinors. The idea of an effective potential that is unbounded from below can be traced back to the classical dynamics, where a charged particle subject to a constant electric field would accelerate without boundary. The Dirac spinors corresponding to equations (\[eige\_16a\]) and (\[eige\_16b\]) are proportional to $$\begin{aligned}
\phi_{D}^{+ \prime} &= e^{i k^{\prime}_{\perp} x^{\prime}_{\perp}} \left( \begin{array}{c}
  b_{+}^{\prime}  D_{-a^{\prime}-1}(\xi^{\prime}) - D_{-a^{\prime}}(\xi^{\prime}) \\
  b_{+}^{\prime}  D_{-a^{\prime}-1}(\xi^{\prime}) + D_{-a^{\prime}}(\xi^{\prime}) \\
\end{array} \right) \, , \label{eige_19a} \\
\phi_{D}^{- \prime} &= e^{i k^{\prime}_{\perp} x^{\prime}_{\perp}} \left( \begin{array}{c}
 D_{-a^{\prime}}(\xi^{\prime}) + b_{-}^{\prime}  D_{-a^{\prime}-1}(\xi^{\prime}) \\
 D_{-a^{\prime}}(\xi^{\prime}) - b_{-}^{\prime}  D_{-a^{\prime}-1}(\xi^{\prime}) \\
\end{array} \right) \, , \label{eige_19b}\end{aligned}$$ with weights $$b_{\pm}^{'} = l^{\prime}_{\mathcal{E}} e^{i \pi /4} (k_{3}^{\prime} \pm i k^{\prime}_{2})/\sqrt{2} \, . \label{eige_20}$$

In the laboratory reference frame, the continuous spectrum is maintained and, up to a global phase factor, the eigenstates are given by $$\begin{aligned}
\label{eige_21t}
\phi_{D}^{+} &=e^{i k_{\perp} x_{\perp}}  \left[ \Xi_{\mathcal{E}}^{+} + \Xi_{\mathcal{E}}^{-} \sigma_{2} \right] \left( \begin{array}{c}
  b_{+}  D_{-a-1}(\xi) - D_{-a}(\xi) \\
  b_{+}  D_{-a-1}(\xi) + D_{-a}(\xi) \\
\end{array} \right)  \, , \label{eige_21a}\\
\phi_{D}^{-} &=e^{i k_{\perp} x_{\perp}}  \left[ 