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

In the framework of the so-called quasi-Hermitian quantum mechanics of stationary unitary systems, bound states are usually constructed as eigenstates |ψ_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 Ω: H → mathfrak{h}. Naturally, the well known ambiguity of these H-dependent Dyson-map transformations implies also an ambiguity of the physical, Ω-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 Ω=Ω(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.