= Solution
Put $\Lambda=|\lambda|\gg E_L$. Changing the sign of $\lambda$ translates the potential by $L/2$ without changing its <energy eigenvalues>, so take a minimum at $x=0$ and expand there. The <parabolic approximation> gives
$$
V(x)\simeq-\Lambda+\frac\Lambda2\left(\frac{2\pi x}{L}\right)^2
=-\Lambda+\frac12m\omega_{\mathrm{loc}}^2x^2,
\qquad \omega_{\mathrm{loc}}=\frac{2\pi}{L}\sqrt{\frac\Lambda m}.
$$
The low-lying <wavefunctions> are localized well inside one period. A local <gauge transformation> removes $a$ from their <quantum harmonic oscillator> equation, giving
$$
\widetilde\epsilon_n(a)\simeq-\Lambda+\hbar\omega_{\mathrm{loc}}\left(n+\frac12\right).
$$
Consequently,
$$
\boxed{\widetilde\epsilon_1(0)-\widetilde\epsilon_0(0)
\simeq\frac{2\pi\hbar}{L}\sqrt{\frac{|\lambda|}{m}}
=\sqrt{2|\lambda|E_L}.}
$$
The sketch consists of two nearly horizontal branches, separated by this gap, as in the right panel above. The <parabolic approximation> makes them completely independent of <magnetic flux>; in the actual ring they retain exponentially small periodic dependence through <quantum tunnelling> around the ring. Corrections from the nonquadratic part of the potential shift the mean energies and gap by relative order $\sqrt{E_L/\Lambda}$, whereas the tunnelling dependence is smaller than every power of that parameter.
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