The energy spectrum of complex periodic potentials of the Kronig-Penney type

We consider a complex periodic PT-symmetric potential of the Kronig-Penney type, in order to elucidate the peculiar properties found by Bender et al. for potentials of the form $V=i(\sin x)^{2N+1}$, and in particular the absence of anti-periodic solutions. In this model we show explicitly why these solutions disappear as soon as $V^*(x)\neq V(x)$, and spell out the consequences for the form of the dispersion relation.