= Solution
In the normalized <plane wave> basis, the only nonzero potential <matrix elements> are
$$
V_{mn}=-\frac\lambda2\bigl(\delta_{m,n+1}+\delta_{m,n-1}\bigr).
$$
Thus a useful small parameter is $|\lambda|/E_L\ll1$, which suppresses admixture of well-separated <energy levels>. <First-order nondegenerate perturbation theory> further requires $|\lambda|/2\ll|E_n-E_{n\pm1}|$ for the coupled states. This latter condition fails near a half-integer $\alpha$ even for very small $\lambda$; there one must use <degenerate perturbation theory> in the nearly degenerate pair. Smallness is measured against the gaps to the other states, not the vanishing gap inside this pair.
Near $\alpha=1/2$, keep $n=0,1$. The resulting <Hamiltonian operator> matrix and its <eigenvalues> are
$$
H_{\mathrm{pair}}=\begin{pmatrix}E_L\alpha^2&-\lambda/2\\-\lambda/2&E_L(1-\alpha)^2\end{pmatrix},
$$
$$
\widetilde\epsilon_{0,1}\simeq E_L\left[(\alpha-1/2)^2+\frac14\right]
\mp\sqrt{E_L^2(\alpha-1/2)^2+\frac{\lambda^2}{4}}.
$$
In particular,
$$
\boxed{\widetilde\epsilon_{0,1}(\pi\hbar/L)
=\frac{\pi^2\hbar^2}{2mL^2}\mp\frac{|\lambda|}{2}
+O(\lambda^2/E_L).}
$$
For positive $\lambda$, the lower <quantum state> at the crossing is the symmetric combination of $n=0,1$, and the upper state is the antisymmetric combination. A negative $\lambda$ exchanges these combinations but leaves the ordered <energy eigenvalues> unchanged.
This <cosine-potential gap in a quantum ring> appears in the middle panel above: each lowest-branch crossing opens into an <avoided crossing> of gap $|\lambda|$ to first order. Away from these crossings, the diagonal <matrix elements> vanish, so the leading shifts are second order. At integer $\alpha$, the two first-excited <plane waves> differ by two units of $n$ and therefore have no first-order coupling; their smaller splitting begins at second order. The numerical plot includes that splitting.
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