Hybrid form of quantum theory with non-Hermitian Hamiltonians

TL;DR

This paper introduces a hybrid approach to non-Hermitian quantum theory that combines operator transformation and metric amendment strategies, demonstrated on a two-state system, to facilitate Hermitization of non-Hermitian Hamiltonians.

Contribution

It presents a novel hybrid Hermitization method combining existing strategies, enhancing the theoretical framework for non-Hermitian quantum systems.

Findings

Hybrid Hermitization simplifies the treatment of non-Hermitian Hamiltonians.

The approach is demonstrated on a schematic two-state quantum system.

It offers a unified framework for operator and metric-based model building.

Abstract

In 1956, Freeman Dyson discovered that the practical solution of Schr\"{o}dinger equation may be facilitated when one replaces the standard self-adjoint Hamiltonian $\mathfrak{h}=\mathfrak{h}^\dagger$ by its manifestly non-Hermitian isospectral avatar $H =\Omega^{-1} \mathfrak{h} \Omega$ with $\Omega^\dagger\Omega=\Theta \neq I$. The standard unitary interpretation of the evolution can be then achieved either by the ``operator transformation'' (OT) reconstruction of $\mathfrak{h}$, or via a metric-amendment (MA) change of the Hilbert space, ${\cal H}_{auxiliary}\to {\cal H}_{physical}$. In our present letter we describe an alternative Hermitization of a ``hybrid form'' (HF) which simply combines the OT model-building strategy with the MA model-building strategy. The merits of the approach are illustrated using a schematic two-state quantum system.