Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 34B / c

For $V=2\lambda \cos (2\pi x/a)$, only $U_{\pm 1}=1$ are nonzero. To first order there is one gap of width $2\lambda$ at the first Brillouin-zone boundary $k=\pm \pi /a$; higher gaps vanish at this order. The allowed bands are the folded free-electron parabolas with that interval removed.

For the Dirac comb, Poisson summation gives $U_l=1$ for every integer $l$. Hence every crossing at $k=l\pi /a$ opens a gap of width $2\lambda$ to first order, producing an infinite sequence of allowed bands between consecutive gaps.