 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 givesThe low-lying wavefunctions are localized well inside one period. A local gauge transformation removes $a$ from their quantum harmonic oscillator equation, givingConsequently,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.