Quasi-Hermitian supersymmetric extensions of a non-Hermitian oscillator Hamiltonian and of its generalizations

TL;DR

This paper explores supersymmetric extensions of non-Hermitian, PT-symmetric harmonic oscillator Hamiltonians using superalgebra, leading to new non-Hermitian Hamiltonians related to Hermitian counterparts.

Contribution

It introduces quasi-Hermitian supersymmetric extensions by enlarging su(1,1) to a superalgebra, enabling the construction of new non-Hermitian Hamiltonians related to Hermitian ones.

Findings

Constructed new non-Hermitian Hamiltonians via superalgebra extension.

Reviewed examples of supersymmetric non-Hermitian Hamiltonians.

Demonstrated relation to Hermitian Hamiltonians through similarity transformations.

Abstract

A harmonic oscillator Hamiltonian augmented by a non-Hermitian \pt-symmetric part and its su(1,1) generalizations, for which a family of positive-definite metric operators was recently constructed, are re-examined in a supersymmetric context. Quasi-Hermitian supersymmetric extensions of such Hamiltonians are proposed by enlarging su(1,1) to a ${\rm su}(1,1/1) \sim {\rm osp}(2/2, \R)$ superalgebra. This allows the construction of new non-Hermitian Hamiltonians related by similarity to Hermitian ones. Some examples of them are reviewed.