Higher Order Integrals Of Motion Research Articles

Higher order superintegrability, Painlevé transcendents and representations of polynomial algebras

Résumé. In recent years, progress toward the classification of superintegrable systems with higher order integrals of motion has been made. In particular, a complete classification of all exotic potentials with a third or a fourth order integrals, and allowing separation of variables in Cartesian coordinates. All doubly exotic potentials with a fifth order integral have also been completely classified. It has been demonstrated how the Chazy class of third order differential equations plays an important role in solving determining equations. Moreover, taking advantage of various operator algebras defined as Abelian, Heisenberg, Conformal and Ladder case of operator algebras, we re-derived these models. These new techniques also provided further examples of superintegrable Hamiltonian with integrals of arbitrary order. It has been conjectured that all quantum superintegrable potentials that do not satisfy any linear equation satisfy nonlinear equations having the Painlevé property. In addition, it has been discovered that their integrals naturally generate finitely generated polynomial algebras and the representations can be exploited to calculate the energy spectrum. For certain very interesting cases associated with exceptional orthogonal polynomials, these algebraic structures do not allow to calculate the full spectrum and degeneracies. It has been demonstrated that alternative sets of integrals which can be build and used to provide a complete solution. This this allow to make another conjecture i.e. that higher order superintegrable systems can be solved algebraically, they require alternative set of integrals than the one provided by a direct approach.

Integrable and Superintegrable Systems with Higher Order Integrals of Motion: Master Function Formalism

We construct two-dimensional integrable and superintegrable systems in terms of the master function formalism and relate them to Mielnik’s and Marquette’s construction in supersymmetric quantum mechanics. For two different cases of the master functions, we obtain two different two-dimensional superintegrable systems with higher order integrals of motion.

Recurrence approach and higher rank cubic algebras for the N-dimensional superintegrable systems

By applying the recurrence approach and coupling constant metamorphosis, we construct higher order integrals of motion for the Stackel equivalents of the N-dimensional superintegrable Kepler–Coulomb model with non-central terms and the double singular oscillators of type (). We show how the integrals of motion generate higher rank cubic algebra with structure constants involving Casimir operators of the Lie algebras L1 and L2. The realizations of the cubic algebras in terms of deformed oscillators enable us to construct finite dimensional unitary representations and derive the degenerate energy spectra of the corresponding superintegrable systems.

New families of superintegrable systems from Hermite and Laguerre exceptional orthogonal polynomials

In recent years, many exceptional orthogonal polynomials (EOP) were introduced and used to construct new families of 1D exactly solvable quantum potentials, some of which are shape invariant. In this paper, we construct from Hermite and Laguerre EOP and their related quantum systems new 2D superintegrable Hamiltonians with higher-order integrals of motion and the polynomial algebras generated by their integrals of motion. We obtain the finite-dimensional unitary representations of the polynomial algebras and the corresponding energy spectrum. We also point out a new type of degeneracies of the energy levels of these systems that is associated with holes in sequences of EOP.

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.

Integrable systems on the sphere associated with genus three algebraic curves

New variables of separation for few integrable systems on the two-dimensional sphere with higher order integrals of motion are considered in detail. We explicitly describe canonical transformations of initial physical variables to the variables of separation and vice versa, calculate the corresponding quadratures and discuss some possible integrable deformations of initial systems.

An infinite family of superintegrable systems from higher order ladder operators and supersymmetry

We discuss how we can obtain new quantum superintegrable Hamiltonians allowing the separation of variables in Cartesian coordinates with higher order integrals of motion from ladder operators. We also discuss how higher order supersymmetric quantum mechanics can be used to obtain systems with higher order ladder operators and their polynomial Heisenberg algebra. We present a new family of superintegrable systems involving the fifth Painlevé transcendent which possess fourth order ladder operators constructed from second order supersymmetric quantum mechanics. We present the polynomial algebra of this family of superintegrable systems.

Construction of classical superintegrable systems with higher order integrals of motion from ladder operators

We construct integrals of motion for multidimensional classical systems from ladder operators of one-dimensional systems. This method can be used to obtain new systems with higher order integrals. We show how these integrals generate a polynomial Poisson algebra. We consider a one-dimensional system with third order ladder operators and found a family of superintegrable systems with higher order integrals of motion. We obtain also the polynomial algebra generated by these integrals. We calculate numerically the trajectories and show that all bounded trajectories are closed.

Supersymmetry as a method of obtaining new superintegrable systems with higher order integrals of motion

The main result of this article is that we show that from supersymmetry we can generate new superintegrable Hamiltonians. We consider a particular case with a third order integral and apply Mielnik’s construction in supersymmetric quantum mechanics. We obtain a new superintegrable potential separable in Cartesian coordinates with a quadratic and quintic integrals and also one with a quadratic integral and an integral of order of 7. We also construct a superintegrable system written in terms of the fourth Painlevé transcendent with a quadratic integral and an integral of order of 7.

Change of the time for the periodic Toda lattices and natural systems on the plane with higher order integrals of motion

We discuss some special classes of canonical transformations of the time variable, which relate different integrable systems. Such dual systems have different integrals of motion, Lax equations, separated variables and bi-hamiltonian structures. As an example the two-dimensional periodic Toda lattices associated with the classical root systems and the dual natural systems on the palne are considered in detail.

Superintegrable systems with third-order integrals of motion

Two-dimensional superintegrable systems with one third-order and one lower order integral of motion are reviewed. The fact that Hamiltonian systems with higher order integrals of motion are not the same in classical and quantum mechanics is stressed. New results on the use of classical and quantum third-order integrals are presented in sections 5 and 6.

Logarithmic conformal field theory solutions of two dimensional magnetohydrodynamics

We consider the application of logarithmic conformal field theory in finding solutions to the turbulent phases of 2-dimensional models of magnetohydrodynamics. These arise upon dimensional reduction of standard (infinite conductivity) 3-dimensional magnetohydrodynamics, after taking various simplifying limits. We show that solutions of the corresponding Hopf equations and higher order integrals of motion can be found within the solutions of ordinary turbulence proposed by Flohr, based on the tensor product of the logarithmic extension c ̃ 6,1 of the non-unitary minimal model c 6,1. This possibility arises because of the existence of a continuous hidden symmetry present in the latter models, and the fact that there appear several distinct dimension-1 and -2 primary fields.