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Three-Hilbert-Space Formulation of Quantum Mechanics
Symmetry, integrability and geometry: methods and applications, Tome 5 (2009) Cet article a éte moissonné depuis la source Math-Net.Ru

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In paper [Znojil M., Phys. Rev. D 78 (2008), 085003, 5 pages, arXiv:0809.2874] the two-Hilbert-space (2HS, a.k.a. cryptohermitian) formulation of Quantum Mechanics has been revisited. In the present continuation of this study (with the spaces in question denoted as $\mathcal H^{\mathrm{ (auxiliary)}}$ and $\mathcal H^{\mathrm{(standard)}}$) we spot a weak point of the 2HS formalism which lies in the double role played by $\mathcal H^{\mathrm{(auxiliary)}}$. As long as this confluence of roles may (and did!) lead to confusion in the literature, we propose an amended, three-Hilbert-space (3HS) reformulation of the same theory. As a byproduct of our analysis of the formalism we offer an amendment of the Dirac's bra-ket notation and we also show how its use clarifies the concept of covariance in time-dependent cases. Via an elementary example we finally explain why in certain quantum systems the generator $H_{\mathrm{(gen)}}$ of the time-evolution of the wave functions may differ from their Hamiltonian $H$.
Keywords: formulation of Quantum Mechanics; cryptohermitian operators of observables; triplet of the representations of the Hilbert space of states; the covariant picture of time evolution. 
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     author = {Miloslav Znojil},
     title = {Three-Hilbert-Space {Formulation} of {Quantum} {Mechanics}},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2009_5_a0/}
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Miloslav Znojil. Three-Hilbert-Space Formulation of Quantum Mechanics. Symmetry, integrability and geometry: methods and applications, Tome 5 (2009). http://geodesic.mathdoc.fr/item/SIGMA_2009_5_a0/

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