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In the recently quickly developing context of quantum mechanics of unitarysystems using a time-independent non-Hermitian Hamiltonian $H$ (having realspectrum and defined as acting in an unphysical but user-friendly Hilbert space${\cal R}_N^{(0)}$), the present paper introduces and describes a set ofconstructive returns of the description to one of the correct and eligiblephysical Hilbert spaces ${\cal R}_0^{(j)}$. The superscript $j$ may run from$j=0$ to $j=N$. In the $j=0$ extreme of the theory the construction iscurrently well known and involves solely the inner product metric$\Theta=\Theta(H)$. The Hamiltonian $H$ itself remains unchanged. At $j=N$ theinner-product metric remains trivial and only the Hamiltonian must beHermitized, $H \to \mathfrak{h} = \Omega\,H\,\Omega^{-1}=\mathfrak{h}^\dagger$.At the remaining superscripts $j=1,2,\ldots, N-1$, a new, hybrid form of theconstruction of a consistent quantum model is proposed, requiring asimultaneous amendment of both the metric and the Hamiltonian. In applications,one of these options is expected to be optimal for a given $H$ in a wayillustrated by a schematic three-state example.

Systematics of quasi-Hermitian representations of non-Hermitian quantum models