Actually, I feel rather retarded for posting this question. Below is the answer.

The equation in question is indeed the Schrödinger's equation for a standard harmonic oscillator. See [this link](http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc2.html).

It can be written as

$$-\frac{1}{2}\frac{\partial^2 \phi_1(x^1)}{(\partial x^1)^2} + \frac{\omega^2}{2} \left(x^1-\frac{k_2}{eB} \right)^2 \phi_1(x^1) = E' \phi_1(x^1),$$

where $\omega = e B$ and $E' = \frac{1}{2}\left(E^2 - (k_3-eA)^2 + eB\right)$.

$\frac{k_2}{eB}$ simply shifts the center of oscillation.

For a harmonic oscillator, we have $E' = \left(\frac{1}{2} + n_1 \right)\omega$. Therefore

$$E^2 = (k_3-eA)^2 + 2 n_1 eB = \left(\frac{2\pi n_3}{L}-eA\right)^2 + 2 n_1 eB.$$

This actually looks better than (19.9) in Zhong-Zhi Xianyu's solution.

However, I have not figured out part (e).

As far as the equation above is concerned, when a certain $n_3$ become invalid as we change background $A$ adiabatically, $-n_3$ at the same time would become valid. ($n_3$ can be either positive or negative integers.)

My guess is that the first-order equations of $\phi_1$ and $\phi_2$ will put further restrictions on what values of $n_3$ are actually allowed.