Apply second-order perturbation theory to a one-dimensional periodic perturbing potential
$$
V(x)=\sum_{n=-\infty}^{+\infty} V_n e^{2 \pi i n x / \xi}
$$
with period $\xi$. To enforce closely spaced discrete energies, assume that the entire "crystal" has length $L=N \xi$ and use the periodic boundary condition $\Psi(x+L / 2)=$ $\Psi(x-L / 2)$, where $N$ is a large even number. Assume that the zero-order Hamiltonian is that of a free particle subject to these boundary conditions. Show that nondegenerate perturbation theory breaks down at the band edges and that the forbidden energy gaps are proportional to the Fourier coefficients of the potential.
Show that in the limit $L \rightarrow \infty$, carried out as in Section 1 of the Appendix, the unperturbed (unnormalized) energy eigenfunctions are plane waves, $e^{i k x}$, and derive the second-order approximation to the dispersion function $E(k)$.