$2\times 2 $ matrix form $$\begin{aligned}
{\mathcal H}=\begin{pmatrix} U & v_F\left(\pi_x-i\pi_y\right) \\
v_F\left(\pi_x+i\pi_y\right) & U \end{pmatrix},\end{aligned}$$ where $v_F$ ($\approx 10^6~{\rm m/s}$) is the graphene Fermi velocity. The $x$ and $y$ components of of the kinetic momentum operator, $\pi_x$ and $\pi_y$, are given by $$\begin{aligned}
\pi_x=\frac{\hbar}{i}\frac{d}{dx},~\pi_y=\hbar q+eA_y,\end{aligned}$$ where $e$ is the elementary charge and $q$ is the $y$ component of the wave vector, which is a constant of the motion. The 2D stationary Dirac equation, which follows from the Hamiltonian, takes the form $$\begin{aligned}
\frac{d}{dx}\begin{pmatrix}\psi_A \\ \psi_B \end{pmatrix}
&&=\begin{pmatrix} ka+q & ik(1-u) \\ ik(1-u) & -ka-q \end{pmatrix}\begin{pmatrix}\psi_A \\ \psi_B \end{pmatrix}\nonumber\\
&&=k\begin{pmatrix} \beta & i\epsilon \\ i\epsilon & -\beta \end{pmatrix}\begin{pmatrix}\psi_A \\ \psi_B \end{pmatrix},
\label{eq:dirac}\end{aligned}$$ where $k$ \[$=E/(\hbar v_F)$\] is the wave number for the electron wave in free space and $E$ ($>0$) is the electron energy. The dimensionless scalar and vector potentials $u$ and $a$ and the supplementary functions $\epsilon$ and $\beta$ are defined by $$\begin{aligned}
u=\frac{U}{E},~a=\frac{eA_y}{\hbar  k},~\epsilon=1-u,~\beta=\frac{q}{k}+a.\end{aligned}$$ It is easy to see that the localization behavior in the negative energy case is the same as that in the positive energy case with the opposite signs of the scalar and vector potentials.

We are interested in the situation where $U$ and $A_y$ 