On the Role of the Normalization Factors $\kappa_n$ and of the Pseudo-Metric $\mathcal P\neq\mathcal P^\dagger$ in Crypto-Hermitian Quantum Models

Among $\mathcal P$-pseudo-Hermitian Hamiltonians $H=\mathcal P^{-1}\,H^\dagger\,\mathcal P$ with real spectra, the "weakly pseudo-Hermitian" ones (i.e. those employing non-self-adjoint $\mathcal P\neq\mathcal P^\dagger$) form a remarkable subfamily. We list some reasons why it deserves a special attention. In particular we show that whenever $\mathcal P\neq\mathcal P^\dagger$, 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 innerproducts in the physical Hilbert space of states, with $\langle\psi_1\,|\,\mathcal{PQ}\,|\,\psi_2\rangle=\langle\psi_1\,|\,\mathcal{CP}\,|\,\psi_2\rangle $.