Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

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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