334 research outputs found
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 pseudo-Hermitian and
symmetric. Specific rotation-like generalized parities are considered
such that 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 point Runge-Kutta version 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
defining its non-equivalent physical Hilbert spaces
of states) is constructed, in closed form, for any coupling and any matrix dimension .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 . 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 which defines the physical Hilbert space of
states can help us to clarify several apparent paradoxes. We argue that with a
suitable 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 is
chosen antilinear. Secondly, both these components of a super-Hamiltonian
are defined along certain topologically nontrivial complex curves
which spread over several Riemann sheets of the wave function.
The non-uniqueness of our choice of the map between "tobogganic"
partner curves and 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 ,
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
- âŠ