= Cosine-potential gap in a quantum ring
In the <plane wave> basis of a ring, $V(x)=-\lambda\cos(2\pi x/L)$ has <matrix elements> $V_{mn}=-(\lambda/2)(\delta_{m,n+1}+\delta_{m,n-1})$. Near $\alpha=1/2$, <degenerate perturbation theory> on $n=0,1$ gives
$$
E_\pm=E_L[(\alpha-1/2)^2+1/4]\pm\sqrt{E_L^2(\alpha-1/2)^2+\lambda^2/4},
$$
up to corrections from remote states. Thus the <avoided crossing> gap is $|\lambda|$ to first order when $|\lambda|\ll E_L$.
For $|\lambda|\gg E_L$, a <parabolic approximation> about a potential minimum instead gives a local <quantum harmonic oscillator> spacing $\sqrt{2|\lambda|E_L}$. A local <gauge transformation> removes the flux from this approximation. Actual ring levels retain exponentially small flux dependence through <quantum tunnelling>, while nonquadratic corrections to their mean energies are algebraic in $\sqrt{E_L/|\lambda|}$.
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