Smorodinsky Winternitz System Research Articles

N-dimensional Smorodinsky–Winternitz model and related higher rank quadratic algebra

The N-dimensional Smorodinsky–Winternitz system is a maximally superintegrable and exactly solvable model, being subject of study from different approaches. The model has been demonstrated to be multiseparable with wavefunctions given by Laguerre and Jacobi polynomials. In this paper we present the complete symmetry algebra of the system, which it is a higher-rank quadratic one containing the recently discovered Racah algebra as subalgebra. The substructures of distinct quadratic algebras and their related Casimirs are also studied. In this way, from the constraints on the oscillator realizations of these substructures, the energy spectrum of the N-dimensional Smorodinsky–Winternitz system is obtained. We show that allows different set of substructures based on the Racah algebra which can be applied independently to algebraically derive the spectrum of the system.

Classical multiseparable Hamiltonian systems, superintegrability and Haantjes geometry

We show that the theory of classical Hamiltonian systems admitting separating variables can be formulated in the context of (ω,H) structures. They are symplectic manifolds endowed with a compatible Haantjes algebra H, namely an algebra of (1,1)tensor fields with vanishing Haantjes torsion. A special class of coordinates, called Darboux-Haantjes coordinates, will be constructed from the Haantjes algebras associated with a separable system. These coordinates enable the additive separation of variables of the corresponding Hamilton-Jacobi equation.We shall prove that a multiseparable system admits as many ωH structures as separation coordinate systems. In particular, we will show that a large class of multiseparable, superintegrable systems, including the Smorodinsky-Winternitz systems and some physically relevant systems with three degrees of freedom, possesses multiple Haantjes structures.

Superintegrability on the three-dimensional spaces with curvature. Oscillator-related and Kepler-related systems on the sphere S 3 and on the hyperbolic space H 3

The superintegrability of several Hamiltonian systems defined on three-dimensional configuration spaces of constant curvature is studied. We first analyze the properties of the Killing vector fields, Noether symmetries and Noether momenta. Then we study the superintegrability of the harmonic oscillator, the Smorodinsky–Winternitz system and the harmonic oscillator with ratio of frequencies 1:1:2 and additional nonlinear terms on the three-dimensional sphere S 3 (κ > 0) and on the hyperbolic space H 3 (κ < 0). In the second part we present a study first of the Kepler problem and then of the Kepler problem with additional nonlinear terms in these two curved spaces, S 3 (κ > 0) and H 3 (κ < 0). We prove their superintegrability and we obtain, in all the cases, the maximal number of functionally independent integrals of motion. All the mathematical expressions are presented using the curvature κ as a parameter, in such a way that particularizing for κ > 0, κ = 0, or κ < 0, the corresponding properties are obtained for the system on the sphere S 3, the Euclidean space , or the hyperbolic space H 3, respectively.

Embedding of the Racah algebra R(n) and superintegrability

The rank-1 Racah algebra R(3) plays a pivotal role in the theory of superintegrable systems. It appears as the symmetry algebra of the 3-parameter system on the 2-sphere from which all second-order conformally flat superintegrable models in 2D can be obtained by means of suitable limits and contractions. A higher rank generalisation of R(3), the so-called rank n−2 Racah algebra R(n), has been considered recently and showed to be the symmetry algebra of the general superintegrable model on the (n−1)-sphere. In the present work, we show that such an algebraic structure naturally arises as embedded inside a larger quadratic algebra characterising nD superintegrable models with non-central terms. This is shown both in classical and quantum mechanics through suitable (symplectic or differential) realisations of the Racah and additional generators. Among the main results, we present an explicit construction of the complete symmetry algebras for two families of n-dimensional maximally superintegrable models, the Smorodinsky–Winternitz system and the generalised Kepler–Coulomb system. For both families, the underlying symmetry algebras are higher-rank quadratic algebras containing the Racah algebra R(n) as subalgebra. High-order algebraic relations among the generators of the full quadratic algebras are also obtained both in the classical and quantum frameworks. These results should shed new light to the further understanding of the structures of quadratic algebras in the context of superintegrable systems.

Quantum SU(2|1) supersymmetric \u2102N Smorodinsky-Winternitz system

We study quantum properties of SU(2|1) supersymmetric (deformed mathcal{N} = 4, d = 1 supersymmetric) extension of the superintegrable Smorodinsky-Winternitz system on a complex Euclidian space ℂN. The full set of wave functions is constructed and the energy spectrum is calculated. It is shown that SU(2|1) supersymmetry implies the bosonic and fermionic states to belong to separate energy levels, thus exhibiting the “even-odd” splitting of the spectra. The superextended hidden symmetry operators are also defined and their action on SU(2|1) multiplets of the wave functions is given. An equivalent description of the same system in terms of superconformal SU(2|1, 1) quantum mechanics is considered and a new representation of the hidden symmetry generators in terms of the SU(2|1, 1) ones is found.

Formula omitted]-Smorodinsky–Winternitz system in a constant magnetic field

We propose the superintegrable generalization of Smorodinsky–Winternitz system on the N-dimensional complex Euclidian space which is specified by the presence of constant magnetic field. We find out that in addition to 2N Liouville integrals the system has additional functionally independent constants of motion, and compute their symmetry algebra. We perform the Kustaanheimo–Stiefel transformation of C2-Smorodinsky–Winternitz system to the (three-dimensional) generalized MICZ-Kepler problem and find the symmetry algebra of the latter one. We observe that constant magnetic field appearing in the initial system has no qualitative impact on the resulting system.

Generalized Lenard chains and multi-separability of the Smorodinsky–Winternitz system

We show that the notion of generalized Lenard chains allows to formulate in a natural way the theory of multi-separable systems in the context of bi-Hamiltonian geometry. We prove that the existence of generalized Lenard chains generated by a Hamiltonian function and by a Nijenhuis tensor defined on a symplectic manifold guarantees the separation of variables. As an application, we construct such a chain for the case I of the classical Smorodinsky–Winternitz model.

From constants of motion to superposition rules for Lie–Hamilton systems

A Lie system is a non-autonomous system of first-order differential equations possessing a superposition rule, i.e. a map expressing its general solution in terms of a generic finite family of particular solutions and some constants. Lie–Hamilton systems form a subclass of Lie systems whose dynamics is governed by a curve in a finite-dimensional real Lie algebra of functions on a Poisson manifold. It is shown that Lie–Hamilton systems are naturally endowed with a Poisson coalgebra structure. This allows us to devise methods for deriving in an algebraic way their constants of motion and superposition rules. We illustrate our methods by studying Kummer–Schwarz equations, Riccati equations, Ermakov systems and Smorodinsky–Winternitz systems with time-dependent frequency.

A new approach to the higher order superintegrability of the Tremblay–Turbiner–Winternitz system

The higher order superintegrability of systems separable in polar coordinates is studied using an approach that was previously applied for the study of the superintegrability of a generalized Smorodinsky–Winternitz system. The idea is that the additional constant of motion can be factorized as the product of powers of two particular rather simple complex functions (here denoted by M and N). This technique leads to a proof of the superintegrability of the Tremblay–Turbiner–Winternitz system and to the explicit expression of the constants of motion. A second family (related with the first one) of superintegrable systems is also studied.

Lie-algebraic description of the quantum superintegrable Smorodinsky–Winternitz system in n dimensions

We apply the potential group method to a family of n-dimensional quantum Smorodinsky–Winternitz systems. The Hamiltonians of the systems are associated with first-order Casimir operators of the unitary group U(3n) restricted to certain subspaces of carrier space of the symmetric representation. Hence, the group U(3n) describes fixed energy states of a family of Smorodinsky–Winternitz systems with different potential strength. Moreover, it is shown that 2n − 1 integrals of motions (including the Hamiltonian) are related to Casimir operators of U(3n) and its subgroups.

Generalized Lenard chains, separation of variables, and superintegrability

We show that the notion of generalized Lenard chains naturally allows formulation of the theory of multiseparable and superintegrable systems in the context of bi-Hamiltonian geometry. We prove that the existence of generalized Lenard chains generated by a Hamiltonian function defined on a four-dimensional ωN manifold guarantees the separation of variables. As an application, we construct such chains for the Hénon-Heiles systems and for the classical Smorodinsky-Winternitz systems. New bi-Hamiltonian structures for the Kepler potential are found.

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.

Revisiting the Symmetries of the Quantum Smorodinsky-Winternitz System in D Dimensions

The $D$-dimensional Smorodinsky-Winternitz system, proposed some years ago by Evans, is re-examined from an algebraic viewpoint. It is shown to possess a potential algebra, as well as a dynamical potential one, in addition to its known symmetry and dynamical algebras. The first two are obtained in hyperspherical coordinates by introducing $D$ auxiliary continuous variables and by reducing a 2D-dimensional harmonic oscillator Hamiltonian. The su(2D) symmetry and ${\rm w}(2D) \oplus_s {\rm sp}(4D,{\mathbb R})$ dynamical algebras of this Hamiltonian are then transformed into the searched for potential and dynamical potential algebras of the Smorodinsky-Winternitz system. The action of generators on wavefunctions is given in explicit form for D=2.

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.

A Super-Integrable Two-Dimensional Non-Linear Oscillator with an Exactly Solvable Quantum Analog

Two super-integrable and super-separable classical systems which can be con- sidered as deformations of the harmonic oscillator and the Smorodinsky-Winternitz in two dimensions are studied and identified with motions in spaces of constant curvature, the deformation parameter being related with the curvature. In this sense these systems are to be considered as a harmonic oscillator and a Smorodinsky-Winternitz system in such bi-dimensional spaces of constant curvature. The quantization of the first system will be carried out and it is shown that it is super-solvable in the sense that the Schrodinger equa- tion reduces, in three different coordinate systems, to two separate equations involving only one degree of freedom.

Maximal superintegrability onN-dimensional curved spaces

A unified algebraic construction of the classical Smorodinsky-Winternitz systems on the ND sphere, Euclidean and hyperbolic spaces through the Lie groups SO(N+1), ISO(N), and SO(N,1) is presented. Firstly, general expressions for the Hamiltonian and its integrals of motion are given in a linear ambient space $R^{N+1}$, and secondly they are expressed in terms of two geodesic coordinate systems on the ND spaces themselves, with an explicit dependence on the curvature as a parameter. On the sphere, the potential is interpreted as a superposition of N+1 oscillators. Furthermore each Lie algebra generator provides an integral of motion and a set of 2N-1 functionally independent ones are explicitly given. In this way the maximal superintegrability of the ND Euclidean Smorodinsky-Winternitz system is shown for any value of the curvature.

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.