On the Role of the Normalization Factors κ n and of the Pseudo - Metric P 6 = P † in Crypto - Hermitian Quantum Models ?

Among P-pseudo-Hermitian Hamiltonians H = P−1H† P with real spectra, the “weakly pseudo-Hermitian” ones (i.e., those employing non-self-adjoint P 6 = P†) form a remarkable subfamily. We list some reasons why it deserves a special attention. In particular we show that whenever P 6 = P†, 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 〈ψ1 | PQ |ψ2〉 = 〈ψ1 | CP |ψ2〉.