Conditional observability versus self-duality in a schematic model

TL;DR

This paper investigates a PT-symmetric model showing how breaking spectral reflection symmetry affects the domain where the Hamiltonian remains quasi-Hermitian with real energies.

Contribution

It demonstrates the connection between spectral symmetry violation and the geometric complexity of the parameter domain in a schematic PT-symmetric model.

Findings

Violation of self-duality correlates with complex boundary shapes of the quasi-Hermitian domain.

The model illustrates how spectral symmetry influences the reality of bound-state energies.

Loss of reflection symmetry leads to a more complicated parameter space boundary.

Abstract

In a simple PT-symmetric model we demonstrate that and how the violation of a reflection symmetry $E_j=-E_{N+1-j}$ of the spectrum (called "self-duality" by Dunne and Shifman) is connected with the loss of the simplicity of the shape of the boundary of the domain D of parameters where the Hamiltonian H is quasi-Hermitian, i.e., where all the bound-state energies are real.