The Floquet Method for $PT$-symmetric Periodic Potentials

TL;DR

This paper clarifies how the Floquet method ensures the secular equation for $PT$-symmetric periodic potentials is real, supporting the real or conjugate pair nature of energy levels in such systems.

Contribution

It demonstrates the reality of the secular equation in the Floquet method for $PT$-symmetric periodic potentials, resolving a key theoretical ambiguity.

Findings

The secular equation in $PT$-symmetric periodic potentials is real.

The reality of the secular equation supports real or conjugate energy levels.

The paper provides a theoretical clarification for the Floquet method in this context.

Abstract

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.