Origin of branch points in the spectrum of PT-symmetric periodic potentials

TL;DR

This paper investigates the origin of branch points in the energy spectrum of PT-symmetric periodic potentials, providing an analytic criterion for when and where eigenvalue bifurcations occur, and clarifying the physical mechanisms behind spectral transitions.

Contribution

It introduces an analytic criterion to predict bifurcation points in PT-symmetric spectra and explains the physical origin of spectral branch points.

Findings

Derived an analytic criterion for bifurcation points.

Clarified the physical origin of spectrum branch points.

Identified conditions for eigenvalue transition from real to complex.

Abstract

There exists multiple branch points in the energy spectrum for some \emph{PT}-symmetric periodic potentials, where the real eigenvalues turn into complex ones. By studying the transmission amplitude for a localized complex potential, we elucidate the physical origin of the breakdown of perturbation method and Born approximation. Mostly importantly, we derive an analytic criteria to determine why, when and where the bifurcation will occur.