Exact Metric Operators as the Ground State functions of the Hermitian Conjugates of a Class of Quasi-Hermitian Hamiltonians

TL;DR

This paper introduces a simple method to find exact metric operators for a class of non-Hermitian, non-PT-symmetric Hamiltonians with zero-energy ground states, simplifying the analysis of such quantum systems.

Contribution

The authors derive closed-form metric operators for a new class of non-Hermitian Hamiltonians, avoiding complex coupled equations and extending the understanding of quasi-Hermitian systems.

Findings

Exact metric operators are obtained for the class of Hamiltonians.

Ground states have zero energy eigenvalues.

The class relates to imaginary magnetic fields and quasi-gauge transformations.

Abstract

We generalized a class of non-Hermitian Hamiltonians which introduced previously by us in such a way in which every member in the class is non-\textit{PT}-symmetric. For every member of the class, the ground state is a constant with zero energy eigen value. Instead of using an infinite set of coupled operator equations to calculate the metric operator we used a simple realization to obtain the class of closed form metric operators corresponding to the class of non-Hermitian and non-\textit{PT}-symmetric Hamiltonians introduced. The trick is that, if $\psi$ is an eigen function of $H$, then $\phi=\eta\psi$ is an eigen function of $H^{\dagger}$ with the same eigen value. Thus, knowing any pair $(\psi ,\phi)$ one can deduce the form of the exact metric operator. We note that, the class of Hamiltonians generalized in this work has the form of that of imaginary magnetic field which can be absorbed by the quasi-gauge transformations represented by metric operators. Accordingly, it is expected that the $Q$ operators will disappear for the whole members in the class in the path integral formulation. However, the detailed analysis of this issue will appear in another work.