An electron of charge $-e$ and mass $m$ is subject to a magnetic field of the form $\mathbf{B}=(0,0, B(y))$, where $B(y)$ is everywhere greater than some positive constant $B_{0}$. In a stationary state of energy $E$, the electron's wavefunction $\Psi$ satisfies

$$-\frac{\hbar^{2}}{2 m}\left(\boldsymbol{\nabla}+\frac{i e}{\hbar} \mathbf{A}\right)^{2} \Psi+\frac{e \hbar}{2 m} \mathbf{B} \cdot \boldsymbol{\sigma} \Psi=E \Psi,$$

where $\mathbf{A}$ is the vector potential and $\sigma_{1}, \sigma_{2}$ and $\sigma_{3}$ are the Pauli matrices.

Assume that the electron is in a spin down state and has no momentum along the $z$-axis. Show that with a suitable choice of gauge, and after separating variables, equation (*) can be reduced to

$$-\frac{d^{2} \chi}{d y^{2}}+(k+a(y))^{2} \chi-b(y) \chi=\epsilon \chi,$$

where $\chi$ depends only on $y, \epsilon$ is a rescaled energy, and $b(y)$ a rescaled magnetic field strength. What is the relationship between $a(y)$ and $b(y)$ ?

Show that $(* *)$ can be factorized in the form $M^{\dagger} M \chi=\epsilon \chi$ where

$$M=\frac{d}{d y}+W(y)$$

for some function $W(y)$, and deduce that $\epsilon$ is non-negative.

Show that zero energy states exist for all $k$ and are therefore infinitely degenerate.