The Floquet Method for $PT$-symmetric Periodic Potentials

By the general theory of $PT$-symmetric quantum systems, their energy levels are either real or occur in complex-conjugate pairs, which implies that the secular equation must be real. However, for periodic potentials it is by no means clear that the secular equation arising in the Floquet method is indeed real, since it involves two linearly independent solutions of the Schr\"odinger equation. In this brief note we elucidate how that reality can be established.