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


Solution

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

 Solution (source code)