QT-Symmetry and Weak Pseudo-Hermiticity

TL;DR

This paper explores the relationship between QT-symmetry and weak pseudo-Hermiticity in non-Hermitian Hamiltonians, revealing conditions under which they are equivalent and providing examples of Hamiltonians with PT-like spectral properties.

Contribution

It clarifies the conditions linking QT-symmetry and weak pseudo-Hermiticity and presents examples of non-PT-symmetric Hamiltonians with PT-like spectral features.

Findings

QT-symmetry is equivalent to Q^{-1}-weak-pseudo-Hermiticity under certain conditions

Not all QT-symmetric Hamiltonians are weak pseudo-Hermitian, showing a nuanced relationship

Some non-PT-symmetric Hamiltonians share spectral properties with PT-symmetric ones

Abstract

For an invertible (bounded) linear operator Q acting in a Hilbert space ${\cal H}$, we consider the consequences of the QT-symmetry of a non-Hermitian Hamiltonian $H:{\cal H}\to{\cal H}$ where T is the time-reversal operator. If H is symmetric in the sense that ${\cal T} H^\dagger {\cal T}=H$, then QT-symmetry is equivalent to Q^{-1}-weak-pseudo-Hermiticity. But in general this equivalence does not hold. We show this using some specific examples. Among these is a large class of non-PT-symmetric Hamiltonians that share the spectral properties of PT-symmetric Hamiltonians.