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

TL;DR

This paper investigates the energy spectrum of complex PT-symmetric Kronig-Penney potentials, explaining the disappearance of anti-periodic solutions when the potential is non-Hermitian and analyzing the resulting dispersion relations.

Contribution

It provides a detailed analysis of the spectral properties of complex periodic potentials of the Kronig-Penney type, highlighting the effects of PT-symmetry breaking on solutions.

Findings

Anti-periodic solutions vanish when the potential is non-Hermitian.

Dispersion relations are significantly affected by PT-symmetry properties.

Explicit connection to previous work on potentials of the form $V=i( ext{sin} x)^{2N+1}$.

Abstract

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.