Zone-face two-plane-wave gap for a cosine potential

For $-V_0\cos Qx$, the Fourier coupling between the two free states at $k=\pm Q/2$ is $-V_0/2$. The split energies are $E_Q/4\mp V_0/2$, with $E_Q={\hslash }^2{Q}^2/(2m)$. The direct face gap is $V_0$, not $2V_0$.