Solution
= Solution
Let $u=V_{10}<0$ and
$$
E_X=\frac{\hbar^2\pi^2}{2m_ea^2}.
$$
At $X=(\pi/a,0)$ the two degenerate <plane waves> have momenta $(\pi/a,0)$ and $(-\pi/a,0)$ and coupling $u$. Thus
$$
\begin{vmatrix}E_X-E&u\\u&E_X-E\end{vmatrix}=0,\qquad
\boxed{E_{X,1}=E_X+u,\quad E_{X,2}=E_X-u.}
$$
Their symmetric and antisymmetric combinations are the lower and upper states respectively. The local <band gap> is $2|u|$.