On the Role of the Normalization Factors κn and of the Pseudo-Metric P in Crypto-Hermitian Quantum Models

Among P−pseudo-Hermitian Hamiltonians H = P HP 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〉. AMS classification 81Q10; 47B50 Acknowledgement Work supported by GAČR, grant Nr. 202/07/1307, Institutional Research Plan AV0Z10480505 and by the MŠMT “Doppler Institute” project Nr. LC06002.