Superintegrability of the Tremblay-Turbiner-Winternitz quantum Hamiltonians on a plane for odd kk

In a recent FTC by Tremblay {\sl et al} (2009 {\sl J. Phys. A: Math. Theor.} {\bf 42} 205206), it has been conjectured that for any integer value of $k$, some novel exactly solvable and integrable quantum Hamiltonian $H_k$ on a plane is superintegrable and that the additional integral of motion is a $2k$th-order differential operator $Y_{2k}$. Here we demonstrate the conjecture for the infinite family of Hamiltonians $H_k$ with odd $k \ge 3$, whose first member corresponds to the three-body Calogero-Marchioro-Wolfes model after elimination of the centre-of-mass motion. Our approach is based on the construction of some $D_{2k}$-extended and invariant Hamiltonian \chh_k, which can be interpreted as a modified boson oscillator Hamiltonian. The latter is then shown to possess a $D_{2k}$-invariant integral of motion \cyy_{2k}, from which $Y_{2k}$ can be obtained by projection in the $D_{2k}$ identity representation space.Comment: 14 pages, no figure; change of title + important addition to sect. 4 + 2 more references + minor modifications; accepted by JPA as an FT

A Chandra Snapshot Survey for 3C Radio Galaxies with redshifts between 0.3 and 0.5

This paper contains an analysis of short Chandra observations of 19 3C sources with redshifts between 0.3 and 0.5 not previously observed in the X-rays. This sample is part of a project to obtain Chandra data for all of the extragalactic sources in the 3C catalogue. Nuclear X-ray intensities as well as any X-ray emission associated with radio jet knots, hotspots or lobes have been measured in 3 energy bands: soft, medium and hard. Standard X-ray spectral analysis for the 4 brightest nuclei has been also performed. X-ray emission was detected for all the nuclei of the radio sources in the current sample with the exception of 3C 435A. There is one compact steep spectrum (CSS) source while all the others are FRII radio galaxies. X-ray emission from two galaxy clusters (3C 19 and 3C 320); from 6 hotspots in 4 radio galaxies (3C 16, 3C 19, 3C 268.2, 3C 313); and extended X-ray emission on kpc scales in 3C 187 and 3C 313, has been detected.Comment: 33 pages, 22 figures, 7 tables, accepted for publication on the ApJ Supplement Series. arXiv admin note: text overlap with arXiv:1210.602

The outer filament of Centaurus A as seen by MUSE

We investigate signatures of a jet-interstellar medium (ISM) interaction using optical integral-field observations of the so-called outer filament near Centaurus A, expanding on previous results obtained on a more limited area. Using the Multi Unit Spectroscopic Explorer (MUSE) on the VLT during science verification, we observed a significant fraction of the brighter emitting gas across the outer filament. The ionized gas shows complex morphology with compact blobs, arc-like structures and diffuse emission. Based on the kinematics, we identified three main components. The more collimated component is oriented along the direction of the radio jet. The other two components exhibit diffuse morphology together with arc-like structures also oriented along the radio jet direction. Furthermore, the ionization level of the gas is found to decrease from the more collimated component to the more diffuse components. The morphology and velocities of the more collimated component confirm our earlier results that the outer filament and the nearby HI cloud are likely partially shaped by the lateral expansion of the jet. The arc-like structures embedded within the two remaining components are the clearest evidence of a smooth jet-ISM interaction along the jet direction. This suggests that, although poorly collimated, the radio jet is still active and has an impact on the surrounding gas. This result indicates that the effect on the ISM of even low-power radio jets should be considered when studying the influence Active Galactic Nuclei can have on their host galaxy.Comment: 5 pages, 3 figures, Accepted for publication by A&

Third order superintegrable systems separating in polar coordinates

A complete classification is presented of quantum and classical superintegrable systems in $E_2$ that allow the separation of variables in polar coordinates and admit an additional integral of motion of order three in the momentum. New quantum superintegrable systems are discovered for which the potential is expressed in terms of the sixth Painlev\'e transcendent or in terms of the Weierstrass elliptic function

Necessary conditions for classical super-integrability of a certain family of potentials in constant curvature spaces

We formulate the necessary conditions for the maximal super-integrability of a certain family of classical potentials defined in the constant curvature two-dimensional spaces. We give examples of homogeneous potentials of degree -2 on $E^2$ as well as their equivalents on $S^2$ and $H^2$ for which these necessary conditions are also sufficient. We show explicit forms of the additional first integrals which always can be chosen polynomial with respect to the momenta and which can be of an arbitrary high degree with respect to the momenta

Infinite families of superintegrable systems separable in subgroup coordinates

A method is presented that makes it possible to embed a subgroup separable superintegrable system into an infinite family of systems that are integrable and exactly-solvable. It is shown that in two dimensional Euclidean or pseudo-Euclidean spaces the method also preserves superintegrability. Two infinite families of classical and quantum superintegrable systems are obtained in two-dimensional pseudo-Euclidean space whose classical trajectories and quantum eigenfunctions are investigated. In particular, the wave-functions are expressed in terms of Laguerre and generalized Bessel polynomials.Comment: 19 pages, 6 figure

Avancée de Diabrotica virgifera virgifera [Coleoptera : Chrysomelidae] dans les champs de maïs au Québec et collecte dans le soja à Ottawa, Ontario

La chrysomèle des racines du maïs de l’Ouest, Diabrotica virgifera virgifera, a été trouvée au Québec en septembre 2000 dans la région de la Montérégie. Ceci constitue une extension vers le nord de son aire de répartition. De plus, à Ottawa, quelques individus se sont développés à partir du soja. Ceci constitue la première mention de développement de cet insecte sur du soja au Canada.The western corn rootworm, Diabrotica virgifera virgifera, was found in the Monteregie region in the province of Quebec in September 2000. This finding constitutes a northern extension of the species distribution. Moreover, at Ottawa, some specimens were found developing from soybean plants. This constitutes the first mention of development of this insect on soybean in Canada

N=2 supersymmetric extension of the Tremblay-Turbiner-Winternitz Hamiltonians on a plane

The family of Tremblay-Turbiner-Winternitz Hamiltonians $H_k$ on a plane, corresponding to any positive real value of $k$, is shown to admit a ${\cal N} = 2$ supersymmetric extension of the same kind as that introduced by Freedman and Mende for the Calogero problem and based on an ${\rm osp}(2/2, \R) \sim {\rm su}(1,1/1)$ superalgebra. The irreducible representations of the latter are characterized by the quantum number specifying the eigenvalues of the first integral of motion $X_k$ of $H_k$. Bases for them are explicitly constructed. The ground state of each supersymmetrized Hamiltonian is shown to belong to an atypical lowest-weight state irreducible representation.Comment: 18 pages, no figur

Superintegrable systems with spin and second-order integrals of motion

We investigate a quantum nonrelativistic system describing the interaction of two particles with spin 1/2 and spin 0, respectively. We assume that the Hamiltonian is rotationally invariant and parity conserving and identify all such systems which allow additional integrals of motion that are second order matrix polynomials in the momenta. These integrals are assumed to be scalars, pseudoscalars, vectors or axial vectors. Among the superintegrable systems obtained, we mention a generalization of the Coulomb potential with scalar potential $V_0=\frac{\alpha}{r}+\frac{3\hbar^2}{8r^2}$ and spin orbital one $V_1=\frac{\hbar}{2r^2}$.Comment: 32 page