Colour valued Scattering Matrices

We describe a general construction principle which allows to add colour values to a coupling constant dependent scattering matrix. As a concrete realization of this mechanism we provide a new type of S-matrix which generalizes the one of affine Toda field theory, being related to a pair of Lie algebras. A characteristic feature of this S-matrix is that in general it violates parity invariance. For particular choices of the two Lie algebras involved this scattering matrix coincides with the one related to the scaling models described by the minimal affine Toda S-matrices and for other choices with the one of the Homogeneous sine-Gordon models with vanishing resonance parameters. We carry out the thermodynamic Bethe ansatz and identify the corresponding ultraviolet effective central charges.Comment: 8 pages Latex, example, comment and reference adde

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

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 $W$, which we illustrate in the case of some particular affine Toda field theories.Comment: 18p., USP-IFQSC/TH/93-0

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

Time-independent approximations for time-dependent optical potentials

We explore the possibility of modifying the Lewis-Riesenfeld method ofin-variants developed originally to find exact solutions for time-dependent quantum me-chanical systems for the situation in which an exact invariant can beconstructed, butthe subsequently resulting time-independent eigenvalue system is not solvable exactly.We propose to carry out this step in an approximate fashion, such as employing stan-dard time-independent perturbation theory or the WKB approximation, and subsequentlyfeeding the resulting approximated expressions back into the time-dependent scheme. Weillustrate the quality of this approach by contrasting an exactly solvable solution to oneobtained with a perturbatively carried out second step for two types of explicitly time-dependent optical potential

n-Extended Lorentzian Kac-Moody algebras

We investigate a class of Kac–Moody algebras previously not considered. We refer to them as n-extended Lorentzian Kac–Moody algebras defined by their Dynkin diagrams through the connection of an An Dynkin diagram to the node corresponding to the affine root. The cases n=1 and n=2 correspond to the well-studied over- and very-extended Kac–Moody algebras, respectively, of which the particular examples of E10 and E11 play a prominent role in string and M-theory. We construct closed generic expressions for their associated roots, fundamental weights and Weyl vectors. We use these quantities to calculate specific constants from which the nodes can be determined that when deleted decompose the n-extended Lorentzian Kac–Moody algebras into simple Lie algebras and Lorentzian Kac–Moody algebra. The signature of these constants also serves to establish whether the algebras possess SO(1, 2) and/or SO(3)-principal subalgebras

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

Anyonic Interpretation of Virasoro Characters and the Thermodynamic Bethe Ansatz

Employing factorized versions of characters as products of quantum dilogarithms corresponding to irreducible representations of the Virasoro algebra, we obtain character formulae which admit an anyonic quasi-particle interpretation in the context of minimal models in conformal field theories. We propose anyonic thermodynamic Bethe ansatz equations, together with their corresponding equation for the Virasoro central charge, on the base of an analysis of the classical limit for the characters and the requirement that the scattering matrices are asymptotically phaseless.Comment: 20 pages (Latex), minor typos corrections and three references adde