arXiv subjects

Ian Marquette

Publications and source records attributed to Ian Marquette.

115 records · Page 7Linked to original sources

arXiv2010

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 the 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 seven. We also construct a superintegrable system written in terms of the fourth Painleve transcendent with a quadratic integral and an integral of order seven.

math-ph↗

arXiv2009

Superintegrability and higher order polynomial algebras II

In an earlier article, we presented a method to obtain integrals of motion and polynomial algebras for a class of two-dimensional superintegrable systems from creation and annihilation operators. We discuss the general case and present its polynomial algebra. We will show how this polynomial algebra can be directly realized as a deformed oscillator algebra. This particular algebraic structure allows to find the unitary representations and the corresponding energy spectrum. We apply this construction to a family of caged anisotropic oscillators. The method can be used to generate new superintegrable systems with higher order integrals. We obtain new superintegrable systems involving the fourth Painleve transcendent and present their integrals of motion and polynomial algebras.

math-ph↗

arXiv2009

Superintegrability and higher order polynomial algebras I

We present a method to obtain higher order integrals and polynomial algebras for two-dimensional superintegrable systems from creation and annihilation operators. All potentials with a second and a third order integrals 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 the creation and annihilation operators. We obtain quintic and seventh order polynomial algebras. We give also for 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.

math-ph↗

arXiv2009

Superintegrability with third order integrals of motion, cubic algebras and supersymmetric quantum mechanics II:Painleve 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.

math-ph↗

arXiv2009

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 integral 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.

math-ph↗

arXiv2008

Superintegrable Systems with a 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.

math-ph↗

arXiv2007

Polynomial Associative Algebras for Quantum Superintegrable Systems with a Third Order Integral of Motion

We consider a superintegrable Hamiltonian system in a two-dimensional space with a scalar potential that allows one quadratic and one cubic integral of motion. We construct the most general associative 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 one of the quantum potentials.

math-ph↗