Geodesic
On the Role of the Normalization Factors $\kappa_n$ and of the Pseudo-Metric $\mathcal P\neq\mathcal P^\dagger$ in Crypto-Hermitian Quantum Models
Symmetry, integrability and geometry: methods and applications, Tome 4 (2008) Cet article a éte moissonné depuis la source Math-Net.Ru

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Among $\mathcal P$-pseudo-Hermitian Hamiltonians $H=\mathcal P^{-1}\,H^\dagger\,\mathcal P$ with real spectra, the "weakly pseudo-Hermitian" ones (i.e. those employing non-self-adjoint $\mathcal P\neq\mathcal P^\dagger$) form a remarkable subfamily. We list some reasons why it deserves a special attention. In particular we show that whenever $\mathcal P\neq\mathcal P^\dagger$, the current involutive operator of charge $\mathcal C$ gets complemented by a nonequivalent alternative involutive quasiparity operator $\mathcal Q$. We show how, in this language, the standard quantum mechanics is restored via the two alternative innerproducts in the physical Hilbert space of states, with $\langle\psi_1\,|\,\mathcal{PQ}\,|\,\psi_2\rangle=\langle\psi_1\,|\,\mathcal{CP}\,|\,\psi_2\rangle $.
Keywords: $\mathcal{PT}$-symmetry; non-self-adjoint pseudo-metrics; $\mathcal{PQ}$-crypto-Hermiticity; $\mathcal{CP}$-crypto-Hermiticity. 
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     author = {Miloslav Znojil},
     title = {On the {Role} of the {Normalization} {Factors} $\kappa_n$ and of the {Pseudo-Metric} $\mathcal P\neq\mathcal P^\dagger$ in {Crypto-Hermitian} {Quantum} {Models}},
     journal = {Symmetry, integrability and geometry: methods and applications},
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}
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Miloslav Znojil. On the Role of the Normalization Factors $\kappa_n$ and of the Pseudo-Metric $\mathcal P\neq\mathcal P^\dagger$ in Crypto-Hermitian Quantum Models. Symmetry, integrability and geometry: methods and applications, Tome 4 (2008). http://geodesic.mathdoc.fr/item/SIGMA_2008_4_a0/

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