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SIGMA 4 (2008), 001, 9 pages arXiv:0710.4432
https://doi.org/10.3842/SIGMA.2008.001
Contribution to the Proceedings of the 3-rd Microconference Analytic and Algebraic Methods III
On the Role of the Normalization Factors κn and of the Pseudo-Metric P ≠ P† in Crypto-Hermitian Quantum Models
Miloslav Znojil
Ústav jaderné fyziky AV CR, 250 68 Rez, Czech Republic
Received November 26, 2007; Published online January 02, 2008
Abstract
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 ≠ P†) form a remarkable subfamily. We
list some reasons why it deserves a special attention. In
particular we show that whenever P ≠ 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 áψ1|PQ|ψ2ñ = áψ1|CP|ψ2ñ.
Key words:
PT-symmetry; non-self-adjoint pseudo-metrics; PQ-crypto-Hermiticity; CP-crypto-Hermiticity.
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