Three-Hilbert-Space Formulation of Quantum Mechanics

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 $H$.