On the Role of the Normalization Factors ${\kappa }_n$ and of the Pseudo-Metric $\mathcal{P}\mathrm{\ne }{\mathcal{P}}^{†}$ in Crypto-Hermitian Quantum Models

Among $\mathcal{P}$-pseudo-Hermitian Hamiltonians $H ={\cal P}^{-1} H^\dagger \cal P}$ with real spectra, the ''weakly pseudo-Hermitian" ones (i.e., those employing non-self-adjoint $\mathcal{P}\mathrm{\ne }{\mathcal{P}}^{†}$) form a remarkable subfamily. We list some reasons why it deserves a special attention. In particular we show that whenever $\mathcal{P}\mathrm{\ne }{\mathcal{P}}^{†}$, the current involutive operator of charge $\mathcal{C}$ gets complemented by a nonequivalent alternative involutive quasiparity operator $\mathcal{Q}$. We show how, in this language, the standard quantum mechanics is restored via the two alternative inner products in the physical Hilbert space of states, with $<\psi_1|{\cal PQ}|\psi_2>=< \psi_1|{\cal CP}|\psi_2>$.