Exactly solvable models with PT-symmetry and with an asymmetric coupling of channels

Bound states generated by K coupled PT-symmetric square wells are studied in a series of models where the Hamiltonians are assumed $R-$pseudo-Hermitian and $R^2-$symmetric. Specific rotation-like generalized parities $R$ are considered such that $R^N=I$ at some integers N. We show that and how our assumptions make the models exactly solvable and quasi-Hermitian. This means that they possess the real spectra as well as the standard probabilistic interpretation.Comment: 22 p., submitted and to be presented, this week, to PHHQP IV Int. Workshop in Stellenbosch (http://academic.sun.ac.za/workshop

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

Perturbation method for non-square Hamiltonians and its application to polynomial oscillators

A remarkable extension of Rayleigh-Schroedinger perturbation method is found. Its (N+q) x (N+1) - dimensional Hamiltonians (as emerging, e.g., during quasi-exact constructions of bound states) are non-square matrices at q > 1. The role of an eigenvalue is played by an energy/coupling q-plet. In all orders, its perturbations are defined via a q-dimensional inversion.Comment: 21 page

Perturbed Poeschl-Teller oscillators

Wave functions and energies are constructed in a short-range Poeschl-Teller well (= negative quadratic secans hyperbolicus) with a quartic perturbation. Within the framework of an innovated, Lanczos-inspired perturbation theory we show that our choice of non-orthogonal basis makes all the corrections given by closed formulae. The first few items are then generated using MAPLE.Comment: 10 pages, Latex, submitted to Physics Letters

Cryptohermitian Hamiltonians on graphs

A family of nonhermitian quantum graphs (exhibiting, presumably, a hidden form of hermiticity) is proposed and studied via their discretization.Comment: 9 pages, 2 figures, the IJTP-special-issue core of talk presented during PHHQP-9 conference (June 21 - 23, 2010, Hangzhou, China, http://www.math.zju.edu.cn/wjd/

Classification of the conditionally observable spectra exhibiting central symmetry

We show how in PT-symmetric 2J-level quantum systems the assumption of an upside-down symmetry (or duality) of their spectra simplifies their classification based on the non-equivalent pairwise mergers of the energy levels.Comment: 10 pp. 3 figure

Strengthened PT-symmetry with P ≠\neq P†^\dagger

Two alternative scenarios are shown possible in Quantum Mechanics working with non-Hermitian $PT-$symmetric form of observables. While, usually, people assume that $P$ is a self-adjoint indefinite metric in Hilbert space (and that their $P-$pseudo-Hermitian Hamiltonians $H$ possess the real spectra etc), we propose to relax the constraint $P=P^\dagger$ as redundant. Non-Hermitian triplet of coupled square wells is chosen for illustration purposes. Its solutions are constructed and the observed degeneracy of their spectrum is attributed to the characteristic nontrivial symmetry $S={P}^{-1} {P}^\dagger \neq I$ of the model $H$. Due to the solvability of the model the determination of the domain where the energies remain real is straightforward. A few remarks on the correct (albeit ambiguous) physical interpretation of the model are added.Comment: 10 pp. + 1 figur

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

The Coulomb - harmonic oscillator correspondence in PT-symmetric quantum mechanics

We show that and how the Coulomb potential can be regularized and solved exactly at the imaginary couplings. The new spectrum of energies is real and bounded as expected, but its explicit form proves totally different from the usual real-coupling case.Comment: Latex, 11 pages, 3 figures, submitted to Phys. Lett.

Quantum toboggans with two branch points

Wave functions describing quantum toboggans with two branch points (QT2) are defined along complex contours of coordinates which spiral around these branch points. The classification of QT2 is found in terms of certain ``winding descriptors" $\varrho$. A mapping $x^{(\varrho)}(s) \to y^{(0)}(s)$ is then presented which rectifies the contours for a subset of the simplest $\varrho=\varrho_0$.Comment: 19pp, 3 figs, presented also as a part of the lecture for QTS-5 conference in Valladolid, Spain, during July 22-28, 2007 (see its webpage http://tristan.fam.cie.uva.es/qts5