Ian Marquette

Superintegrability with third order integrals of motion, cubic algebras, and supersymmetric quantum mechanics. II. Painlevé transcendent potentials

We consider a superintegrable quantum potential in two-dimensional Euclidean space with a second and a third order integral of motion. The potential is written in terms of the fourth Painleve transcendent. We construct for this system a cubic algebra of integrals of motion. The algebra is realized in terms of parafermionic operators and we present Fock-type representations which yield the corresponding energy spectra. We also discuss this potential from the point of view of higher order supersymmetric quantum mechanics and obtain ground state wave functions.

Superintegrability and higher order polynomial algebras

We present a method to obtain higher order integrals and polynomial algebras for two-dimensional quantum superintegrable systems separable in Cartesian coordinates from ladder operators. All systems with a second- and a third-order integral of motion separable in Cartesian coordinates were studied. The integrals of motion of two of them do not generate a cubic algebra. We construct for these Hamiltonians a higher order polynomial algebra from their ladder operators. We obtain quintic and seventh-order polynomial algebras. We also give for the polynomial algebras of order 7 realizations in terms of deformed oscillator algebras. These realizations and finite-dimensional unitary representations allow us to obtain the energy spectrum. We also apply the construction to the caged anisotropic harmonic oscillator and a system involving the fourth Painleve transcendent.

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 Section 5 and 6.

Superintegrability with third order integrals of motion, cubic algebras, and supersymmetric quantum mechanics. I. Rational function potentials

We consider a superintegrable Hamiltonian system in a two-dimensional space with a scalar potential that allows one quadratic and one cubic integrals of motion. We construct the most general cubic algebra and we present specific realizations. We use them to calculate the energy spectrum. All classical and quantum superintegrable potentials separable in Cartesian coordinates with a third order integral are known. The general formalism is applied to quantum reducible and irreducible rational potentials separable in Cartesian coordinates in E2. We also discuss these potentials from the point of view of supersymmetric and PT-symmetric quantum mechanics.

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 Painleve transcendent with a quadratic integral and an integral of order of 7.

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.

Two-step rational extensions of the harmonic oscillator: exceptional orthogonal polynomials and ladder operators

The type III Hermite Xm exceptional orthogonal polynomial family is generalized to a double-indexed one (with m1 even and m2 odd such that m2 > m1) and the corresponding rational extensions of the harmonic oscillator are constructed by using second-order supersymmetric quantum mechanics. The new polynomials are proved to be expressible in terms of mixed products of Hermite and pseudo-Hermite ones, while some of the associated potentials are linked with rational solutions of the Painlev? IV equation. A novel set of ladder operators for the extended oscillators is also built and shown to satisfy a polynomial Heisenberg algebra of order m2 ? m1 + 1, which may alternatively be interpreted in terms of a special type of (m2 ? m1 + 2)th-order shape invariance property.

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 Painleve 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.

Polynomial Poisson algebras for classical superintegrable systems with a third-order integral of motion

We present polynomial Poisson algebras for the eight classical potentials in two-dimensional Euclidian space that separate in Cartesian coordinates and allow a third-order integral of motion. Some of the classical superintegrable potentials do not coincide with quantum ones, but are their singular limits. We show that all bounded trajectories in these potentials are periodic.

Generalized MICZ-Kepler system, duality, polynomial and deformed oscillator algebras

We present the quadratic algebra of the generalized MICZ-Kepler system in three-dimensional Euclidean space E3 and its dual, the four-dimensional singular oscillator, in four-dimensional Euclidean space E4. We present their realization in terms of a deformed oscillator algebra using the Daskaloyannis construction. The structure constants are, in these cases, functions not only of the Hamiltonian but also of other integrals commuting with all generators of the quadratic algebra. We also present a new algebraic derivation of the energy spectrum of the MICZ-Kepler system on the three sphere S3 using a quadratic algebra. These results point out also that results and explicit formula for structure functions obtained for quadratic, cubic, and higher order polynomial algebras in the context of two-dimensional superintegrable systems may be applied to superintegrable systems in higher dimensions with and without monopoles.