Supersymmetric $\eta$ operators

TL;DR

This paper explores the structure of metric operators in supersymmetric quantum mechanics, demonstrating how they relate to Hermitian Hamiltonians and their superpartners, revealing new insights into non-Hermitian systems.

Contribution

It introduces a method to determine metric operators and non-Hermitian Hamiltonians using supersymmetric structures, extending understanding of their interrelations.

Findings

Metric η operators are unitarily equivalent to Hermitian Hamiltonians with supersymmetry.

Fixing a superpartner Hamiltonian allows determination of the metric and non-Hermitian Hamiltonian.

Additional restrictions can relate non-Hermitian Hamiltonians to other Hermitian superpartners.

Abstract

Being chosen as a differential operator of a special form, metric $\eta$ operator becomes unitary equivalent to a one-dimensional Hermitian Hamiltonian with a natural supersymmetric structure. We show that fixing the superpartner of this Hamiltonian permits to determine both the metric operator and corresponding non-Hermitian Hamiltonian. Moreover, under an additional restriction on the non-Hermitian Hamiltonian, it becomes a superpartner of another Hermitian Hamiltonian.