Nearly-free electron dispersion near a one-dimensional band gap
= Nearly-free electron dispersion near a one-dimensional band gap
For
$$
V(x)=\sum_{n\in\mathbb Z}V_ne^{2\pi inx/a},
$$
write $k=n\pi/a+\kappa$. Near the $n$th Bragg crossing,
$$
E_\pm(k)=V_0+\frac{\hbar^2}{2m}
\left[\left(\frac{n\pi}{a}\right)^2+\kappa^2\right]
\pm\sqrt{
\left(\frac{\hbar^2n\pi\kappa}{ma}\right)^2+|V_n|^2}.
$$
At $\kappa=0$, the two branches differ by $2|V_n|$.