Superintegrable systems with spin and second-order integrals of motion

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.

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Carter's constant and superintegrability

International Journal of Modern Physics D ◽

10.1142/s0218271818500669 ◽

2018 ◽

Vol 27 (07) ◽

pp. 1850066

Author(s):

Payel Mukhopadhyay ◽

K. Rajesh Nayak

Keyword(s):

Dynamical System ◽

Degrees Of Freedom ◽

Three Dimensional ◽

Two Dimensional ◽

Integrals Of Motion ◽

Superintegrable System ◽

Simple Observation ◽

Superintegrable Systems ◽

Axisymmetric Spacetime ◽

Three Degrees Of Freedom

Carter's constant is a nontrivial conserved quantity of motion of a particle moving in stationary axisymmetric spacetime. In the version of the theorem originally given by Carter, due to the presence of two Killing vectors, the system effectively has two degrees of freedom. We propose an extension to the first version of Carter's theorem to a system having three degrees of freedom to find two functionally independent Carter-like integrals of motion. We further generalize the theorem to a dynamical system with [Formula: see text] degrees of freedom. We further study the implications of Carter's constant to superintegrability and present a different approach to probe a superintegrable system. Our formalism gives another viewpoint to a superintegrable system using the simple observation of separable Hamiltonian according to Carter's criteria. We then give some examples by constructing some two-dimensional superintegrable systems based on this idea and also show that all three-dimensional simple classical superintegrable potentials are also Carter separable.

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Superintegrable Systems with Algebraic and Rational Integrals of Motion

Theoretical and Mathematical Physics ◽

10.1134/s0040577919050040 ◽

2019 ◽

Vol 199 (2) ◽

pp. 659-674 ◽

Cited By ~ 3

Author(s):

A. V. Tsiganov

Keyword(s):

Integrals Of Motion ◽

Superintegrable Systems

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Dirac equation with anisotropic oscillator, quantum E3′ and Holt superintegrable potentials and relativistic generalized Yang–Coulomb monopole system

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887817500049 ◽

2016 ◽

Vol 14 (01) ◽

pp. 1750004 ◽

Cited By ~ 2

Author(s):

Vahid Mohammadi ◽

Alireza Chenaghlou

Keyword(s):

Dirac Equation ◽

Relativistic Energy ◽

Integrals Of Motion ◽

Superintegrable System ◽

Energy Eigenvalues ◽

System A ◽

Superintegrable Systems ◽

Casimir Operators ◽

Oscillator Quantum ◽

Anisotropic Oscillator

The two-dimensional Dirac equation with spin and pseudo-spin symmetries is investigated in the presence of the maximally superintegrable potentials. The integrals of motion and the quadratic algebras of the superintegrable quantum [Formula: see text], anisotropic oscillator and the Holt potentials are studied. The corresponding Casimir operators and the structure functions of the mentioned superintegrable systems are found. Also, we obtain the relativistic energy spectra of the corresponding superintegrable systems. Finally, the relativistic energy eigenvalues of the generalized Yang–Coulomb monopole (YCM) superintegrable system (a [Formula: see text] non-Abelian monopole) are calculated by the energy spectrum of the eight-dimensional oscillator which is dual to the former system by Hurwitz transformation.

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Quadratic algebra contractions and second-order superintegrable systems

Analysis and Applications ◽

10.1142/s0219530514500377 ◽

2014 ◽

Vol 12 (05) ◽

pp. 583-612 ◽

Cited By ~ 5

Author(s):

Ernest G. Kalnins ◽

W. Miller

Keyword(s):

Constant Curvature ◽

Two Dimensions ◽

Second Order ◽

Quadratic Algebra ◽

Quadratic Algebras ◽

Physical Systems ◽

Special Cases ◽

Superintegrable Systems ◽

Wilson Polynomials ◽

Symmetry Algebras

Quadratic algebras are generalizations of Lie algebras; they include the symmetry algebras of second-order superintegrable systems in two dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical and quantum mechanics. For constant curvature spaces, we show that the free quadratic algebras generated by the first- and second-order elements in the enveloping algebras of their Euclidean and orthogonal symmetry algebras correspond one-to-one with the possible superintegrable systems with potential defined on these spaces. We describe a contraction theory for quadratic algebras and show that for constant curvature superintegrable systems, ordinary Lie algebra contractions induce contractions of the quadratic algebras of the superintegrable systems that correspond to geometrical pointwise limits of the physical systems. One consequence is that by contracting function space realizations of representations of the generic superintegrable quantum system on the 2-sphere (which give the structure equations for Racah/Wilson polynomials) to the other superintegrable systems one obtains the full Askey scheme of orthogonal hypergeometric polynomials.

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