Families of superintegrable Hamiltonians constructed from exceptional polynomials

We introduce a family of exactly-solvable two-dimensional Hamiltonians whose wave functions are given in terms of Laguerre and exceptional Jacobi polynomials. The Hamiltonians contain purely quantum terms which vanish in the classical limit leaving only a previously known family of superintegrable systems. Additional, higher-order integrals of motion are constructed from ladder operators for the considered orthogonal polynomials proving the quantum system to be superintegrable

Deformed oscillator algebras for two dimensional quantum superintegrable systems

Quantum superintegrable systems in two dimensions are obtained from their classical counterparts, the quantum integrals of motion being obtained from the corresponding classical integrals by a symmetrization procedure. For each quantum superintegrable systema deformed oscillator algebra, characterized by a structure function specific for each system, is constructed, the generators of the algebra being functions of the quantum integrals of motion. The energy eigenvalues corresponding to a state with finite dimensional degeneracy can then be obtained in an economical way from solving a system of two equations satisfied by the structure function, the results being in agreement to the ones obtained from the solution of the relevant Schrodinger equation. The method shows how quantum algebraic techniques can simplify the study of quantum superintegrable systems, especially in two dimensions.Comment: 22 pages, THES-TP 10/93, hep-the/yymmnn

Path Integral Approach for Superintegrable Potentials on Spaces of Non-constant Curvature: II. Darboux Spaces DIII and DIV

This is the second paper on the path integral approach of superintegrable systems on Darboux spaces, spaces of non-constant curvature. We analyze in the spaces \DIII and \DIV five respectively four superintegrable potentials, which were first given by Kalnins et al. We are able to evaluate the path integral in most of the separating coordinate systems, leading to expressions for the Green functions, the discrete and continuous wave-functions, and the discrete energy-spectra. In some cases, however, the discrete spectrum cannot be stated explicitly, because it is determined by a higher order polynomial equation. We show that also the free motion in Darboux space of type III can contain bound states, provided the boundary conditions are appropriate. We state the energy spectrum and the wave-functions, respectively

Superintegrable potentials on 3D Riemannian and Lorentzian spaces with non-constant curvature

A quantum sl(2,R) coalgebra is shown to underly the construction of a large class of superintegrable potentials on 3D curved spaces, that include the non-constant curvature analogues of the spherical, hyperbolic and (anti-)de Sitter spaces. The connection and curvature tensors for these "deformed" spaces are fully studied by working on two different phase spaces. The former directly comes from a 3D symplectic realization of the deformed coalgebra, while the latter is obtained through a map leading to a spherical-type phase space. In this framework, the non-deformed limit is identified with the flat contraction leading to the Euclidean and Minkowskian spaces/potentials. The resulting Hamiltonians always admit, at least, three functionally independent constants of motion coming from the coalgebra structure. Furthermore, the intrinsic oscillator and Kepler potentials on such Riemannian and Lorentzian spaces of non-constant curvature are identified, and several examples of them are explicitly presented.Comment: 14 pages. Based in the contribution presented at the Group 27 conference, Yerevan, Armenia, August 13-19, 200

Higher Order Quantum Superintegrability: a new "Painlev\'e conjecture"

We review recent results on superintegrable quantum systems in a two-dimensional Euclidean space with the following properties. They are integrable because they allow the separation of variables in Cartesian coordinates and hence allow a specific integral of motion that is a second order polynomial in the momenta. Moreover, they are superintegrable because they allow an additional integral of order $N>2$. Two types of such superintegrable potentials exist. The first type consists of "standard potentials" that satisfy linear differential equations. The second type consists of "exotic potentials" that satisfy nonlinear equations. For $N= 3$, 4 and 5 these equations have the Painlev\'e property. We conjecture that this is true for all $N\geq3$. The two integrals X and Y commute with the Hamiltonian, but not with each other. Together they generate a polynomial algebra (for any $N$) of integrals of motion. We show how this algebra can be used to calculate the energy spectrum and the wave functions.Comment: 23 pages, submitted as a contribution to the monographic volume "Integrability, Supersymmetry and Coherent States", a volume in honour of Professor V\'eronique Hussin. arXiv admin note: text overlap with arXiv:1703.0975

Massless geodesics in AdS5×Y(p,q)AdS_5\times Y(p,q) as a superintegrable system

A Carter like constant for the geodesic motion in the $Y(p,q)$ Einstein-Sasaki geometries is presented. This constant is functionally independent with respect to the five known constants for the geometry. Since the geometry is five dimensional and the number of independent constants of motion is at least six, the geodesic equations are superintegrable. We point out that this result applies to the configuration of massless geodesic in $AdS_5\times Y(p,q)$ studied by Benvenuti and Kruczenski, which are matched to long BPS operators in the dual N=1 supersymmetric gauge theory.Comment: 20 pages, no figures. Small misprint is corrected in the Killing-Yano tensor. No change in any result or conclusion

Different experimental approaches in modelling cataractogenesis: An overview of selenite-induced nuclear cataract in rats

Cataract, the opacification of eye lens, is the leading cause of blindness worldwide. At present, the only remedy is surgical removal of the cataractous lens and substitution with a lens made of synthetic polymers. However, besides significant costs of operation and possible complications, an artificial lens just does not have the overall optical qualities of a normal one. Hence it remains a significant public health problem, and biochemical solutions or pharmacological interventions that will maintain the transparency of the lens are highly required. Naturally, there is a persistent demand for suitable biological models. The ocular lens would appear to be an ideal organ for maintaining culture conditions because of lacking blood vessels and nerves. The lens in vivo obtains its nutrients and eliminates waste products via diffusion with the surrounding fluids. Lens opacification observed in vivo can be mimicked in vitro by addition of the cataractogenic agent sodium selenite (Na2SeO3) to the culture medium. Moreover, since an overdose of sodium selenite induces also cataract in young rats, it became an extremely rapid and convenient model of nuclear cataract in vivo. The main focus of this review will be on selenium (Se) and its salt sodium selenite, their toxicological characteristics and safety data in relevance of modelling cataractogenesis, either under in vivo or in vitro conditions. The studies revealing the mechanisms of lens opacification induced by selenite are highlighted, the representatives from screening for potential anti-cataract agents are listed

The development of compositions and methods of tool steels heat-treatment with microalloying complex, designed for heavy mechanical engineering products

On the basis of a statistical regression analysis of experimental data for the manufacture of large diameter rolls have been recommended three grades of steel, with a high complex of properties: 100H3G2MTR, 70H3G2FTR and 70H3G2VTB. The features of the microstructure formation experienced in steels according to different heat treatment regimes. The effect of microalloying complex phase transformations in steel during heating and cooling. Designed hardening heat treatment mode, providing improved operational stability of the tool roll.На основе регрессионного статистического анализа экспериментальных данных для изготовления валков большого диаметра были рекомендованы три марки стали, обладающие высоким комплексом свойств: 100Х3Г2МТР, 70Х3Г2ФТР и 70Х3Г2ВТБ. Изучены особенности формирования микроструктуры опытных сталей в зависимости от различных режимов термической обработки. Определено влияние микролегирующего комплекса на фазовые превращения в стали при нагреве и охлаждении. Разработаны упрочняющие режим термической обработки, обеспечивающие повышение эксплуатационной стойкости валкового инструмента