= Solution
Normalize the <plane waves> in a periodic crystal of area or volume $\mathcal V$. The <Fourier coefficients> of the <periodic potential> obey
$$
\langle\mathbf k|V|\mathbf k'\rangle=\frac1{\mathcal V}\int_{\mathcal V}V(\mathbf r)e^{-i(\mathbf k-\mathbf k')\cdot\mathbf r}\,d\mathbf r=V_{\mathbf k-\mathbf k'}.
$$
Translation of the integral by any <lattice vector> $\mathbf R$ leaves it invariant but multiplies it by $e^{-i(\mathbf k-\mathbf k')\cdot\mathbf R}$. A nonzero result therefore requires $\mathbf k-\mathbf k'$ to be a <reciprocal lattice vector>.
Near an isolated <Bragg scattering> degeneracy, ordinary nondegenerate <perturbation theory> fails because the free energies coincide. Retain the resonant <plane waves> $|\mathbf k\rangle$, $|\mathbf k+\mathbf G\rangle$ and apply <degenerate perturbation theory>. Absorb the constant $V_{\mathbf0}$ into the energy origin. Since a real potential has $V_{-\mathbf G}=V_{\mathbf G}^*$, the projected <Schrödinger equation> has coefficient matrix
$$
\begin{pmatrix}
\epsilon_0(\mathbf k)-E&V_{-\mathbf G}\\
V_{\mathbf G}&\epsilon_0(\mathbf k+\mathbf G)-E
\end{pmatrix}.
$$
A nonzero eigenstate requires
$$
\boxed{[\epsilon_0(\mathbf k)-E][\epsilon_0(\mathbf k+\mathbf G)-E]-|V_{\mathbf G}|^2=0.}
$$
Off-resonant <plane waves> give higher-order energy shifts omitted in this leading <nearly-free electron model>.
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