94 research outputs found
PT-symmetric Deformations of the Korteweg-de Vries Equation
We propose a new family of complex PT-symmetric extensions of the Korteweg-de
Vries equation. The deformed equations can be associated to a sequence of
non-Hermitian Hamiltonians. The first charges related to the conservation of
mass, momentum and energy are constructed. We investigate solitary wave
solutions of the equation of motion for various boundary conditions.Comment: 11 pages, 3 figure
PT-symmetry and Integrability
We briefly explain some simple arguments based on pseudo Hermiticity,
supersymmetry and PT-symmetry which explain the reality of the spectrum of some
non-Hermitian Hamiltonians. Subsequently we employ PT-symmetry as a guiding
principle to construct deformations of some integrable systems, the
Calogero-Moser-Sutherland model and the Korteweg deVries equation. Some
properties of these models are discussed.Comment: Proceeding of the Micro conference Analytic and algebraic methods II,
Doppler Institute, Prague, April 200
The two dimensional harmonic oscillator on a noncommutative space with minimal uncertainties
The two dimensional set of canonical relations giving rise to minimal
uncertainties previously constructed from a q-deformed oscillator algebra is
further investigated. We provide a representation for this algebra in terms of
a flat noncommutative space and employ it to study the eigenvalue spectrum for
the harmonic oscillator on this space. The perturbative expression for the
eigenenergy indicates that the model might possess an exceptional point at
which the spectrum becomes complex and its PT-symmetry is spontaneously broken.Comment: 4 pages, contribution to proceedings of "Analytic and algebraic
methods in physics X", Pragu
Comment on "Non-Hermitian Quantum Mechanics with Minimal Length Uncertainty"
We demonstrate that the recent paper by Jana and Roy entitled ''Non-Hermitian
quantum mechanics with minimal length uncertainty'' [SIGMA 5 (2009), 083, 7
pages, arXiv:0908.1755] contains various misconceptions. We compare with an
analysis on the same topic carried out previously in our manuscript
[arXiv:0907.5354]. In particular, we show that the metric operators computed
for the deformed non-Hermitian Swanson models differs in both cases and is
inconsistent in the former
Chaos in the thermodynamic Bethe ansatz
We investigate the discretized version of the thermodynamic Bethe ansatz
equation for a variety of 1+1 dimensional quantum field theories. By computing
Lyapunov exponents we establish that many systems of this type exhibit chaotic
behaviour, in the sense that their orbits through fixed points are extremely
sensitive with regard to the initial conditions.Comment: 10 pages, Late
Minimal areas from q-deformed oscillator algebras
We demonstrate that dynamical noncommutative space-time will give rise to
deformed oscillator algebras. In turn, starting from some q-deformations of
these algebras in a two dimensional space for which the entire deformed Fock
space can be constructed explicitly, we derive the commutation relations for
the dynamical variables in noncommutative space-time. We compute minimal areas
resulting from these relations, i.e. finitely extended regions for which it is
impossible to resolve any substructure in form of measurable knowledge. The
size of the regions we find is determined by the noncommutative constant and
the deformation parameter q. Any object in this type of space-time structure
has to be of membrane type or in certain limits of string type.Comment: 14 pages, 1 figur
A note on the integrability of non-Hermitian extensions of Calogero-Moser-Sutherland models
We consider non-Hermitian but PT-symmetric extensions of Calogero models,
which have been proposed by Basu-Mallick and Kundu for two types of Lie
algebras. We address the question of whether these extensions are meaningful
for all remaining Lie algebras (Coxeter groups) and if in addition one may
extend the models beyond the rational case to trigonometric, hyperbolic and
elliptic models. We find that all these new models remain integrable, albeit
for the non-rational potentials one requires additional terms in the extension
in order to compensate for the breaking of integrability.Comment: 10 pages, Late
Metrics and isospectral partners for the most generic cubic PT-symmetric non-Hermitian Hamiltonian
We investigate properties of the most general PT-symmetric non-Hermitian
Hamiltonian of cubic order in the annihilation and creation operators as a ten
parameter family. For various choices of the parameters we systematically
construct an exact expression for a metric operator and an isospectral
Hermitian counterpart in the same similarity class by exploiting the
isomorphism between operator and Moyal products. We elaborate on the subtleties
of this approach. For special choices of the ten parameters the Hamiltonian
reduces to various models previously studied, such as to the complex cubic
potential, the so-called Swanson Hamiltonian or the transformed version of the
from below unbounded quartic -x^4-potential. In addition, it also reduces to
various models not considered in the present context, namely the single site
lattice Reggeon model and a transformed version of the massive sextic
x^6-potential, which plays an important role as a toy modelto identify theories
with vanishing cosmological constant.Comment: 21 page
Factorized Scattering in the Presence of Reflecting Boundaries
We formulate a general set of consistency requirements, which are expected to
be satisfied by the scattering matrices in the presence of reflecting
boundaries. In particular we derive an equivalent to the boostrap equation
involving the W-matrix, which encodes the reflection of a particle off a wall.
This set of equations is sufficient to derive explicit formulas for , which
we illustrate in the case of some particular affine Toda field theories.Comment: 18p., USP-IFQSC/TH/93-0
Isospectral Hamiltonians from Moyal products
Recently Scholtz and Geyer proposed a very efficient method to compute metric
operators for non-Hermitian Hamiltonians from Moyal products. We develop these
ideas further and suggest to use a more symmetrical definition for the Moyal
products, because they lead to simpler differential equations. In addition, we
demonstrate how to use this approach to determine the Hermitian counterpart for
a Pseudo-Hermitian Hamiltonian. We illustrate our suggestions with the
explicitly solvable example of the -x^4-potential and the ubiquitous harmonic
oscillator in a complex cubic potential.Comment: 10 pages, to appear special issue Czech. J. Phy
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