Third-order Integrals Of Motion Research Articles

Superintegrability with second- and third-order integrals of motion

Superintegrable systems in two-dimensional Euclidean space E 2 with scalar potentials and integrals of motion that are second- or third-order polynomials in the momenta are discussed. We review old results, obtained in collaboration with Ya.A. Smorodinsky more than 40 years ago and present new ones that are their continuation.

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 Painlevé 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 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.

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.

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.

Hamiltonians separable in Cartesian coordinates and third-order integrals of motion

We present in this article all Hamiltonian systems in E(2) that are separable in Cartesian coordinates and that admit a third-order integral, both in quantum and in classical mechanics. Many of these superintegrable systems are new, and it is seen that there exists a relation between quantum superintegrable potentials, invariant solutions of the Korteweg–de Vries equation and the Painlevé transcendents.

Superintegrability with third-order integrals in quantum and classical mechanics

We consider here the coexistence of first- and third-order integrals of motion in two-dimensional classical and quantum mechanics. We find explicitly all potentials that admit such integrals, and all their integrals. Quantum superintegrable systems are found that have no classical analog, i.e., the potentials are proportional to ℏ2, so their classical limit is free motion.