Self-isospectral tri-supersymmetry in PT-symmetric quantum systems with pure imaginary periodicity

TL;DR

This paper explores a PT-symmetric quantum system with complexified potentials, revealing a novel tri-supersymmetric structure that explains its reflectionless and finite-gap properties, and clarifies the state splitting into bound and resonance families.

Contribution

It introduces a new exotic supersymmetric framework based on three matrix differential operators for PT-symmetric systems with pure imaginary periodicity.

Findings

Identifies a tri-supersymmetric structure in the system.

Shows the reflectionless (finite-gap) nature of the potentials.

Explains the splitting of states into bound and resonance families.

Abstract

We study a reflectionless PT-symmetric quantum system described by the pair of complexified Scarf II potentials mutually displaced in the half of their pure imaginary period. Analyzing the rich set of intertwining discrete symmetries of the pair, we find an exotic supersymmetric structure based on three matrix differential operators that encode all the properties of the system, including its reflectionless (finite-gap) nature. The structure we revealed particularly sheds new light on the splitting of the discrete states into two families, related to the bound and resonance states in Hermitian Scarf II counterpart systems, on which two different series of irreducible representations of sl(2,C) are realized.