\begin{array}{c}  \Xi_{\mathcal{B}}^{-} H_{\vert n \vert +1} (z) -
\Xi_{\mathcal{B}}^{+} c_{\pm} H_{\vert n \vert} (z)  \\
i \Xi_{\mathcal{B}}^{-} c_{\pm} H_{\vert n \vert} (z) - i \Xi_{\mathcal{B}}^{+} H_{\vert n \vert +1 } (z)  \\
\end{array} \right) , \label{eige_11b}\end{aligned}$$ with the displaced variable $$\begin{aligned}
\label{eige_12}
z = (1 & -\beta_{\mathcal{B}}^{2})^{1/4} \left(x_{1}/l_{\mathcal{B}} + l_{\mathcal{B}} k_{2} \right) \nonumber \\ &\pm \beta_{\mathcal{B}} \sqrt{2 \vert n \vert + 2 +  \left(1-\beta^{2}_{\mathcal{B}} \right)^{-3/2} (l_{\mathcal{B}} k_{3})^{2}} ,\end{aligned}$$ and relative weights given by transforming equation (\[eige\_9\]) using equation (\[eige\_10\]) and defining the Lorentz parameter $$\label{eige_13}
\Xi^{\pm}_{\mathcal{B}} = \sqrt{1 \pm \left[1 - \beta_{\mathcal{B}}^{2} \right]^{1/2}}  \, .$$ For a given $k_{2}$ and $k_{3}$, the energies of the particle display a $\sqrt{n}$-dependence that is maintained as we transform between reference frames; the separation between adjacent energies is, however, scaled by a factor accounting for dilation or contraction of the magnetic length.

Above critical drive: Continuous spectrum
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The same procedure can be used for $\vert E \vert \geq \vert B \vert$, where a boost in the $\mathbf{e}_{2}$ direction with parameter $$\label{eige_14}
\beta_{\mathcal{E}} = \frac{B}{E} \, ,$$ takes us to a reference frame where the magnetic field vanishes. The corresponding Dirac-Pauli equation then divides into two blocks that lead to the uncoupled equations $$\label{eige_15}
