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

In the recently quickly developing context of quantum mechanics of unitary systems using a time-independent non-Hermitian Hamiltonian $H$ (having real spectrum and defined as acting in an unphysical but user-friendly Hilbert space ${\mathcal{R}}_N^{(0)}$), the present paper introduces and describes a set of constructive returns of the description to one of the correct and eligible physical Hilbert spaces ${\mathcal{R}}_0^{(j)}$. The superscript $j$ may run from $j=0$ to $j=N$. In the $j=0$ extreme of the theory the construction is currently well known and involves solely the inner product metric $\mathrm{\Theta }=\mathrm{\Theta }(H)$. The Hamiltonian $H$ itself remains unchanged. At $j=N$ the inner-product metric remains trivial and only the Hamiltonian must be Hermitized, $H\to \mathfrak{h}=\mathrm{\Omega }H{\mathrm{\Omega }}^{-1}={\mathfrak{h}}^{†}$. At the remaining superscripts $j=1,2,\dots ,N-1$, a new, hybrid form of the construction of a consistent quantum model is proposed, requiring a simultaneous 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 way illustrated by a schematic three-state example.