Twin Hamiltonians, three types of the Dyson maps, and the probabilistic interpretation problem in quasi-Hermitian quantum mechanics

TL;DR

This paper classifies all possible Dyson maps in quasi-Hermitian quantum mechanics, addressing the ambiguity in probabilistic interpretation by systematically exploring transformations linking non-Hermitian Hamiltonians to their Hermitian counterparts.

Contribution

It provides a comprehensive classification of $H$-dependent Dyson maps, offering a systematic framework for defining probabilistic interpretations in quasi-Hermitian quantum systems.

Findings

Classified all eligible $H$-dependent Dyson maps.

Established a systematic framework for probabilistic interpretation.

Addressed ambiguity in non-Hermitian to Hermitian transformations.

Abstract

In the framework of the so-called quasi-Hermitian quantum mechanics of stationary unitary systems, bound states are usually constructed as eigenstates $|\psi_n \rangle$ of a Hamiltonian operator $H$ with real spectrum which is non-Hermitian, $H \neq H^\dagger$. One of the ways of the standard probabilistic interpretation of such systems consists in a transformation of $H$ into its isospectral Hermitian ``twin" $\mathfrak{h}= \mathfrak{h}^\dagger$ via one of the so-called Dyson maps $\Omega: H \to \mathfrak{h}$. Naturally, the well known ambiguity of these $H-$dependent Dyson-map transformations implies also an ambiguity of the physical, $\Omega-$dependent probabilistic and experimental interpretation of the system in question. In the present paper, an exhaustive classification of all of the eligible $H-$dependent Dyson maps $\Omega=\Omega(H)$ is provided, implying also a systematic framework for a specification of all of the possible probabilistic interpretations of the quantum system characterized by a preselected $H$.