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 $\mathrm{\mid }{\psi }_n⟩$ of a Hamiltonian operator $H$ with real spectrum which is non-Hermitian, $H\mathrm{\ne }{H}^{†}$. One of the ways of the standard probabilistic interpretation of such systems consists in a transformation of $H$ into its isospectral Hermitian ``twin" $\mathfrak{h}={\mathfrak{h}}^{†}$ via one of the so-called Dyson maps $\mathrm{\Omega }:H\to \mathfrak{h}$. Naturally, the well known ambiguity of these $H-$dependent Dyson-map transformations implies also an ambiguity of the physical, $\mathrm{\Omega }-$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 $\mathrm{\Omega }=\mathrm{\Omega }(H)$ 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 $H$.