17 research outputs found
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
Third order superintegrable systems separating in polar coordinates
A complete classification is presented of quantum and classical
superintegrable systems in that allow the separation of variables in
polar coordinates and admit an additional integral of motion of order three in
the momentum. New quantum superintegrable systems are discovered for which the
potential is expressed in terms of the sixth Painlev\'e transcendent or in
terms of the Weierstrass elliptic function
Addition theorems and the Drach superintegrable systems
We propose new construction of the polynomial integrals of motion related to
the addition theorems. As an example we reconstruct Drach systems and get some
new two-dimensional superintegrable Stackel systems with third, fifth and
seventh order integrals of motion.Comment: 18 pages, the talk given on the conference "Superintegrable Systems
in Classical and Quantum Mechanics", Prague 200
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.Comment: To appear in J. Phys A: Mathematical and Theoretical (SPE QTS5
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
Generalized five-dimensional Kepler system, Yang-Coulomb monopole, and Hurwitz transformation
The 5D Kepler system possesses many interesting properties. This system is superintegrable and also with a su(2) non-Abelian monopole interaction (Yang-Coulomb monopole). This system is also related to an 8D isotropic harmonic oscillator by a Hurwitz transformation. We introduce a new superintegrable Hamiltonian that consists in a 5D Kepler system with new terms of Smorodinsky-Winternitz type. We obtain the integrals of motion of this system. They generate a quadratic algebra with structure constants involving the Casimir operator of a so(4) Lie algebra. We also show that this system remains superintegrable with a su(2) non-Abelian monopole (generalized Yang-Coulomb monopole). We study this system using parabolic coordinates and obtain from Hurwitz transformation its dual that is an 8D singular oscillator. This 8D singular oscillator is also a new superintegrable system and multiseparable. We obtained its quadratic algebra that involves two Casimir operators of so(4) Lie algebras. This correspondence is used to obtain algebraically the energy spectrum of the generalized Yang-Coulomb monopole
An infinite family of superintegrable systems from higher order ladder operators and supersymmetry
We will 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 will discuss also how higher
order supersymmetric quantum mechanics can be used to obtain systems with
higher order ladder operators and their polynomial Heisenberg algebra. We will
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.Comment: 8 pages, presented at ICGTMP 28, accepted for j.conf.serie
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.Comment: 11 page
Superintegrable systems with spin and second-order integrals of motion
We investigate a quantum nonrelativistic system describing the interaction of
two particles with spin 1/2 and spin 0, respectively. We assume that the
Hamiltonian is rotationally invariant and parity conserving and identify all
such systems which allow additional integrals of motion that are second order
matrix polynomials in the momenta. These integrals are assumed to be scalars,
pseudoscalars, vectors or axial vectors. Among the superintegrable systems
obtained, we mention a generalization of the Coulomb potential with scalar
potential and spin orbital one
.Comment: 32 page
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 . 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 , 4
and 5 these equations have the Painlev\'e property. We conjecture that this is
true for all . The two integrals X and Y commute with the Hamiltonian,
but not with each other. Together they generate a polynomial algebra (for any
) 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