Complete Set of Inner Products for a Discrete PT-symmetric Square-well Hamiltonian

A discrete $N-$point Runge-Kutta version $H^{(N)}({\lambda})$ of one of the simplest non-Hermitian square-well Hamiltonians with real spectrum is studied. A complete set of its possible hermitizations (i.e., of the eligible metrics $\Theta^{(N)}({\lambda})$ defining its non-equivalent physical Hilbert spaces of states) is constructed, in closed form, for any coupling ${\lambda}\in (-1,1)$ and any matrix dimension $N$.Comment: 26 pp., 6 figure

Scattering theory with localized non-Hermiticities

In the context of the recent interest in solvable models of scattering mediated by non-Hermitian Hamiltonians (cf. H. F. Jones, Phys. Rev. D 76, 125003 (2007)) we show that and how the well known variability of our ad hoc choice of the metric $\Theta$ which defines the physical Hilbert space of states can help us to clarify several apparent paradoxes. We argue that with a suitable $\Theta$ a fully plausible physical picture of the scattering is recovered. Quantitatively, our new recipe is illustrated on an exactly solvable toy model.Comment: 22 pp, grammar amende

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 ${\cal H}^{\rm (auxiliary)}$ and ${\cal H}^{\rm (standard)}$) we spot a weak point of the 2HS formalism which lies in the double role played by ${\cal H}^{\rm (auxiliary)}$. 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 $H_{\rm (gen)}$ of the time-evolution of the wave functions may differ from their Hamiltonian $H$

Solvable PT-symmetric model with a tunable interspersion of non-merging levels

We study the spectrum in such a PT-symmetric square well of a diameter L where the "strength of the non-Hermiticity" is controlled by the two parameters, viz., by an imaginary coupling ig and by the distance d of its onset from the origin. We solve this problem and confirm that the spectrum is discrete and real in a non-empty interval of g. Surprisingly, a specific distinction between the bound states is found in their asymptotic stability/instability with respect to an unlimited growth of g. In our model, all of the low-lying levels remain asymptotically unstable at the small d and finite L while only the stable levels survive for d near L or in the purely imaginary well with infinite L. In between these two extremes, an unusual and tunable, variable pattern of the interspersed "robust" and "fragile" subspectra of the real levels is obtained.Comment: final version: 33 pages (plus the old 8 figures of version 1