Non-Hermitian oscillator Hamiltonian and su(1,1): a way towards generalizations

TL;DR

This paper re-examines metric operators for non-Hermitian oscillator Hamiltonians using su(1,1) algebra, proposing an alternative derivation that enables the construction of generalized non-Hermitian oscillators beyond PT-symmetry.

Contribution

It introduces an su(1,1)-based method for deriving metric operators, allowing for the creation of generalized non-Hermitian oscillator Hamiltonians related to Hermitian ones.

Findings

Alternative su(1,1)-based derivation of metric operators

Construction of generalized non-Hermitian oscillator Hamiltonians

Examples demonstrating the approach

Abstract

The family of metric operators, constructed by Musumbu {\sl et al} (2007 {\sl J. Phys. A: Math. Theor.} {\bf 40} F75), for a harmonic oscillator Hamiltonian augmented by a non-Hermitian $\cal PT$-symmetric part, is re-examined in the light of an su(1,1) approach. An alternative derivation, only relying on properties of su(1,1) generators, is proposed. Being independent of the realization considered for the latter, it opens the way towards the construction of generalized non-Hermitian (not necessarily $\cal PT$-symmetric) oscillator Hamiltonians related by similarity to Hermitian ones. Some examples of them are reviewed.