Smorodinsky Winternitz Research Articles

Master symmetries, non-Hamiltonian symmetries and superintegrability of the generalized Smorodinsky–Winternitz system

The first part of the paper is devoted to the theory of master symmetries using the geometric formalism as an approach. It is shown that certain superintegrable systems are endowed with this property as a consequence of the existence of a family of master symmetries. In the second part, the properties of dynamical but non-Hamiltonian symmetries are studied. It is proved that the higher order superintegrability of the generalized Smorodinsky–Winternitz system is a consequence of the existence of symplectic symmetries not preserving the Hamiltonian function.

Two-dimensional generalized harmonic oscillators and their Darboux partners

We construct two-dimensional Darboux partners of the shifted harmonic oscillator potential and of an isotonic oscillator potential belonging to the Smorodinsky–Winternitz class of superintegrable systems. The transformed solutions, their potentials and the corresponding discrete energy spectra are computed in explicit form.

Quadratic algebra approach to relativistic quantum Smorodinsky–Winternitz systems

There exists a relation between the Klein–Gordon and the Dirac equations with scalar and vector potentials of equal magnitude and the Schrödinger equation. We obtain the relativistic energy spectrum for the four relativistic quantum Smorodinsky–Winternitz systems from their quasi-Hamiltonian and the quadratic algebras studied by Daskaloyannis in the nonrelativistic context. We also apply the quadratic algebra approach directly to the initial Dirac equation for these four systems and show that the quadratic algebras obtained are the same than those obtained from the quasi-Hamiltonians. We point out how results obtained in context of quantum superintegrable systems and their polynomial algebras can be applied to the quantum relativistic case.

Nonlinear dynamical symmetries of Smorodinsky—Winternitz and Fokas—Lagerstorm systems

General solutions of the Smorodinsky—Winternitz system and the Fokas—Lagerstorm system, which are superintegrable in two-dimensional Euclidean space, are obtained using the algebraic method (structure function). Their dynamical symmetries, which are governed by deformed angular momentum algebras, are revealed.

Superintegrability on N-dimensional spaces of constant curvature from so(N + 1) and its contractions

The Lie—Poisson algebra so(N + 1) and some of its contractions are used to construct a family of superintegrable Hamiltonians on the N-dimensional spherical, Euclidean, hyperbolic, Minkowskian, and (anti-)de Sitter spaces. We firstly present a Hamiltonian which is a superposition of an arbitrary central potential with N arbitrary centrifugal terms. Such a system is quasi-maximally superintegrable since this is endowed with 2N — 3 functionally independent constants of motion (plus the Hamiltonian). Secondly, we identify two maximally superintegrable Hamiltonians by choosing a specific central potential and finding at the same time the remaining integral. The former is the generalization of the Smorodinsky—Winternitz system to the above six spaces, while the latter is a generalization of the Kepler—Coulomb potential, for which the Laplace—Runge—Lenz N vector is also given. All the systems and constants of motion are explicitly expressed in a unified form in terms of ambient and polar coordinates as they are parametrized by two contraction parameters (curvature and signature of the metric).

A maximally superintegrable system on an [formula omitted]-dimensional space of nonconstant curvature

We present a novel Hamiltonian system in n dimensions which admits the maximal number 2 n − 1 of functionally independent, quadratic first integrals. This system turns out to be the first example of a maximally superintegrable Hamiltonian on an n -dimensional Riemannian space of nonconstant curvature, and it can be interpreted as the intrinsic Smorodinsky–Winternitz system on such a space. Moreover, we provide three different complete sets of integrals in involution and solve the equations of motion in closed form.

SUPERINTEGRABLE SYSTEMS, MULTI-HAMILTONIAN STRUCTURES AND NAMBU MECHANICS IN AN ARBITRARY DIMENSION

A general algebraic condition for the functional independence of 2n-1 constants of motion of an n-dimensional maximal superintegrable Hamiltonian system has been proved for an arbitrary finite n. This makes it possible to construct, in a well-defined generic way, a normalized Nambu bracket which produces the correct Hamiltonian time evolution. Existence and explicit forms of pairwise compatible multi-Hamiltonian structures for any maximal superintegrable system have been established. The Calogero–Moser system, motion of a charged particle in a uniform perpendicular magnetic field and Smorodinsky–Winternitz potentials are considered as illustrative applications and their symmetry algebras as well as their Nambu formulations and alternative Poisson structures are presented.

Two families of superintegrable and isospectral potentials in two dimensions

As an extension of the intertwining operator idea, an algebraic method which provides a link between supersymmetric quantum mechanics and quantum (super)integrability is introduced. By realization of the method in two dimensions, two infinite families of superintegrable and isospectral stationary potentials are generated. The method makes it possible to perform Darboux transformations in such a way that, in addition to the isospectral property, they acquire the superintegrability preserving property. Symmetry generators are second and fourth order in derivatives and all potentials are isospectral with one of the Smorodinsky–Winternitz potentials. Explicit expressions of the potentials, their dynamical symmetry generators, and the algebra they obey as well as their degenerate spectra and corresponding normalizable states are presented.

Algebraic treatment of super-integrable potentials

The so(2,1) Lie algebra is applied to three classes of two- and three-dimensional Smorodinsky–Winternitz super-integrable potentials for which the path integral discussion has been recently presented in the literature. We have constructed the Green’s functions for two important super-integrable potentials in R2. Among the super-integrable potentials in R3, we have considered two examples, one is maximally super-integrable and another one minimally super-integrable. The discussion is made in various coordinate systems. The energy spectrum and the suitably normalized wave functions of bound and continuous states are then deduced.

Dynamical symmetries, bi-Hamiltonian structures, and superintegrable n=2 systems

The theory of dynamical but non-Cartan (or non-Noether) symmetries and the existence of bi-Hamiltonian structures is studied using the symplectic formalism approach. The results are applied to the study of superintegrable systems. It is shown that certain families of n=2 superintegrable systems related with the harmonic oscillator (as, e.g., the so-called Smorodinsky–Winternitz system) are bi-Hamiltonian systems endowed with dynamical symmetries of non-Cartan class.

Group theory of the Smorodinsky–Winternitz system

The three degrees of freedom Smorodinsky–Winternitz system is a degenerate or super-integrable Hamiltonian that possesses five functionally independent globally defined and single-valued integrals of the motion in both classical and quantum mechanics. This is explained in terms of a forced degeneracy imposed as a consequence of the invariance of the Hamiltonian under a group of symmetry transformations isomorphic to the three-dimensional unitary unimodular group, SU(3). In turn, this degeneracy group is embedded in a larger group of transformations that maps all the bound energy levels among each other, the so-called dynamical group. All the bound state eigenfunctions act as basis functions for a single irreducible representation of the dynamical group. So, in common with the hydrogen atom and the harmonic oscillator, the quantum mechanics of the Smorodinsky–Winternitz system may be completely solved within the framework of group theory alone.