Spectral singularities in PT-symmetric periodic finite-gap systems

TL;DR

This paper investigates the origin of spectral singularities in PT-symmetric periodic quantum systems, showing they arise from band-edge state limits and relate to exceptional points and hidden supersymmetry.

Contribution

It reveals the mechanism of spectral singularities emergence in finite-gap PT-symmetric systems and links them to topological changes and nonlinear supersymmetry.

Findings

Spectral singularities emerge from band-edge state limits as the imaginary period tends to infinity.

Spectral singularities are analogous to zero-width resonances in non-periodic systems.

Spectral singularities transform into exceptional points under topological changes.

Abstract

The origin of spectral singularities in finite-gap singly periodic PT-symmetric quantum systems is investigated. We show that they emerge from a limit of band-edge states in a doubly periodic finite gap system when the imaginary period tends to infinity. In this limit, the energy gaps are contracted and disappear, every pair of band states of the same periodicity at the edges of a gap coalesces and transforms into a singlet state in the continuum. As a result, these spectral singularities turn out to be analogous to those in the non-periodic systems, where they appear as zero-width resonances. Under the change of topology from a non-compact into a compact one, spectral singularities in the class of periodic systems we study are transformed into exceptional points. The specific degeneration related to the presence of finite number of spectral singularities and exceptional points is shown to be coherently reflected by a hidden, bosonized nonlinear supersymmetry.