On Pseudo-Hermitian Hamiltonians

TL;DR

This paper explores the construction and non-uniqueness of metric operators for pseudo-Hermitian Hamiltonians, providing methods to generate multiple $\\eta$ operators and analyzing their stability under perturbations.

Contribution

It introduces a systematic way to generate multiple $\\eta$ operators and demonstrates their non-uniqueness for pseudo-Hermitian Hamiltonians.

Findings

A sufficient condition for generating a sequence of $\\eta$ operators.

Proof of non-uniqueness of $\\eta$ operators for a given Hamiltonian.

Analysis of $\\eta$ stability under Hamiltonian perturbations.

Abstract

We investigate some questions on the construction of $\eta$ operators for pseudo-Hermitian Hamiltonians. We give a sufficient condition which can be exploited to systematically generate a sequence of $\eta$ operators starting from a known one, thereby proving the non-uniqueness of $\eta$ for a particular pseudo-Hermitian Hamiltonian. We also study perturbed Hamiltonians for which $\eta$'s corresponding to the original Hamiltonian still work.