On the Role of the Normalization Factors k(n) and of the Pseudo-Metric P not equal P-dagger in Crypto-Hermitian Quantum Models

Among P-pseudo-Hermitian Hamiltonians H = P-1 H-dagger P with real spectra, the weakly pseudo-Hermitian ones (i.e., those employing non-self-adjoint P not equal P-dagger) form a remarkable subfamily. We list some reasons why it deserves a special attention. In particular we show that whenever P not equal P-dagger, the current involutive operator of charge C gets complemented by a nonequivalent alternative involutive quasiparity operator 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 = .