= Solution
Let $G_n=2\pi n/a$. The free states $|k\rangle$ and $|k-G_n\rangle$ are degenerate at the Bragg point $k=G_n/2=n\pi/a$. Since
$$
\langle k|V|k-G_n\rangle=V_n,
$$
<degenerate perturbation theory> restricts the Hamiltonian near this crossing to
$$
\begin{pmatrix}
\hbar^2k^2/(2m)+V_0&V_n\\
V_n^*&\hbar^2(k-G_n)^2/(2m)+V_0
\end{pmatrix}.
$$
Writing $k=n\pi/a+\kappa$ and diagonalizing gives the <nearly-free electron dispersion near a one-dimensional band gap>
$$
\boxed{
E_\pm(k)=V_0+\frac{\hbar^2}{2m}
\left[\left(\frac{n\pi}{a}\right)^2+\kappa^2\right]
\pm\sqrt{
\left(\frac{\hbar^2n\pi\kappa}{ma}\right)^2+|V_n|^2}}.
$$
Thus the two continuous <energy bands>[bands] avoid crossing, and at $\kappa=0$ their separation is
$$
\boxed{\Delta E_n=2|V_n|}.
$$
This is the <nearly-free electron model>. The <dispersion relation> is the relation $E=E(k)$ between energy and Bloch wavevector.
For the specified potential,
$$
\frac83V_0\cos^4\left(\frac{2\pi x}{a}\right)
=V_0+\frac43V_0\cos\left(\frac{4\pi x}{a}\right)
+\frac13V_0\cos\left(\frac{8\pi x}{a}\right).
$$
Hence
$$
V_{\pm2}=\frac23V_0,\qquad
V_{\pm4}=\frac16V_0,
$$
and all other nonconstant Fourier coefficients vanish. The gaps are therefore
$$
\begin{array}{c|c|c}
n&\text{crossing wavevectors}&\text{gap width}\\ \hline
2&k=\pm 2\pi/a&4|V_0|/3\\
4&k=\pm 4\pi/a&|V_0|/3.
\end{array}
$$
Their centre energies are respectively
$$
V_0+\frac{2\hbar^2\pi^2}{ma^2},
\qquad
V_0+\frac{8\hbar^2\pi^2}{ma^2}.
$$
The extended-zone sketch consists of the free-particle parabolas shifted upward by $V_0$, with <avoided crossings> of these two widths at the listed <Bragg points>; elsewhere they meet to this order because the corresponding <Fourier coefficients> vanish.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ii/paper-2-nearly-free-electron-bands.png]
{title=Extended-zone nearly-free-electron bands and their two nonzero gaps}
{description=Gray curves are the intersecting free-electron parabolas. Colored avoided crossings show the larger n equals 2 gap at ka over pi equals plus or minus 2 and the smaller n equals 4 gap at plus or minus 4.}
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