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

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/

CPT-symmetric discrete square well

A new version of an elementary PT-symmetric square well quantum model is proposed in which a certain Hermiticity-violating end-point interaction leaves the spectrum real in a large domain of couplings $\lambda\in (-1,1)$. Within this interval we employ the usual coupling-independent operator P of parity and construct, in a systematic Runge-Kutta discrete approximation, a coupling-dependent operator of charge C which enables us to classify our P-asymmetric model as CPT-symmetric or, equivalently, hiddenly Hermitian alias cryptohermitian.Comment: 12 pp., presented to conference PHHQP IX (http://www.math.zju.edu.cn/wjd/

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

Supersymmetric quantum mechanics living on topologically nontrivial Riemann surfaces

Supersymmetric quantum mechanics is constructed in a new non-Hermitian representation. Firstly, the map between the partner operators $H^{(\pm)}$ is chosen antilinear. Secondly, both these components of a super-Hamiltonian ${\cal H}$ are defined along certain topologically nontrivial complex curves $r^{(\pm)}(x)$ which spread over several Riemann sheets of the wave function. The non-uniqueness of our choice of the map ${\cal T}$ between "tobogganic" partner curves $r^{(+)}(x)$ and $r^{(-)}(x)$ is emphasized.Comment: 14p

Quantum catastrophes: a case study

The bound-state spectrum of a Hamiltonian H is assumed real in a non-empty domain D of physical values of parameters. This means that for these parameters, H may be called crypto-Hermitian, i.e., made Hermitian via an {\it ad hoc} choice of the inner product in the physical Hilbert space of quantum bound states (i.e., via an {\it ad hoc} construction of the so called metric). The name of quantum catastrophe is then assigned to the N-tuple-exceptional-point crossing, i.e., to the scenario in which we leave domain D along such a path that at the boundary of D, an N-plet of bound state energies degenerates and, subsequently, complexifies. At any fixed $N \geq 2$, this process is simulated via an N by N benchmark effective matrix Hamiltonian H. Finally, it is being assigned such a closed-form metric which is made unique via an N-extrapolation-friendliness requirement.Comment: 23 p

Thermodynamics of Pseudo-Hermitian Systems in Equilibrium

In study of pseudo(quasi)-hermitian operators, the key role is played by the positive-definite metric operator. It enables physical interpretation of the considered systems. In the article, we study the pseudo-hermitian systems with constant number of particles in equilibrium. We show that the explicit knowledge of the metric operator is not essential for study of thermodynamic properties of the system. We introduce a simple example where the physically relevant quantities are derived without explicit calculation of either metric operator or spectrum of the Hamiltonian.Comment: 9 pages, 2 figures, to appear in Mod.Phys.Lett. A; historical part of sec. 2.1 reformulated, references corrected; typos correcte

Solvable relativistic quantum dots with vibrational spectra

For Klein-Gordon equation a consistent physical interpretation of wave functions is reviewed as based on a proper modification of the scalar product in Hilbert space. Bound states are then studied in a deep-square-well model where spectrum is roughly equidistant and where a fine-tuning of the levels is mediated by PT-symmetric interactions composed of imaginary delta functions which mimic creation/annihilation processes.Comment: Int. Worskhop "Pseudo-Hermitian Hamiltonians in Quantum Physics III" (June 20 - 22, 2005, Koc Unversity, Istanbul(http://home.ku.edu.tr/~amostafazadeh/workshop/workshop.htm) a part of talk (9 pages

Matching method and exact solvability of discrete PT-symmetric square wells

Discrete PT-symmetric square wells are studied. Their wave functions are found proportional to classical Tshebyshev polynomials of complex argument. The compact secular equations for energies are derived giving the real spectra in certain intervals of non-Hermiticity strengths Z. It is amusing to notice that although the known square well re-emerges in the usual continuum limit, a twice as rich, upside-down symmetric spectrum is exhibited by all its present discretized predecessors.Comment: 25 pp, 3 figure