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With length measured in units of $a=1$, plot the real functionOnly the portions with $|f(k)|\le 1$ can equal $\cos \varphi$, so the strip between the horizontal lines $+1$ and $-1$ identifies allowed energy bands. The band edges of a repulsive delta-comb potential have the following approximate first intervals:Each interval corresponds to energy $E={\hslash }^2{k}^2/(2m)$; the intervening intervals are forbidden. The exact upper endpoints follow from $f(j\pi )=(-1{)}^j$. The lower endpoints solve $f(k)=(-1{)}^{j-1}$ inside $((j-1)\pi ,j\pi )$. The divergence suggested by $\sin k/k$ at the origin is removable, and $f(0)=11>1$, so there is no zero-energy state in the repulsive model.

The periodic barriers produce separated energy bands and band gaps, unlike the continuous free-particle spectrum. A finite $N$-atom loop has discrete states within these bands. Whether the material conducts still depends on band filling: one Electron per primitive cell half-fills a spin-degenerate band, while two Electrons per cell can fill the first band and place the Fermi energy in a gap. Band gaps alone do not establish that every filling is insulating. At high $k$, the perturbing term $10\sin k/k$ becomes relatively small and the spectrum approaches free-electron behaviour in relative terms.