Superintegrable Systems Research Articles

New classes of quadratically integrable systems in magnetic fields: The generalized cylindrical and spherical cases

We study integrable and superintegrable systems with magnetic field possessing quadratic integrals of motion on the three-dimensional Euclidean space. In contrast with the case without vector potential, the corresponding integrals may no longer be connected to separation of variables in the Hamilton–Jacobi equation and can have more general leading order terms. We focus on two cases extending the physically relevant cylindrical- and spherical-type integrals. We find three new integrable systems in the generalized cylindrical case but none in the spherical one. We conjecture that this is related to the presence, respectively absence, of maximal abelian Lie subalgebra of the three-dimensional Euclidean algebra generated by first order integrals in the limit of vanishing magnetic field. By investigating superintegrability, we find only one (minimally) superintegrable system among the integrable ones. It does not separate in any orthogonal coordinate system. This system provides a mathematical model of a helical undulator placed in an infinite solenoid.

Higher-order superintegrable momentum-dependent Hamiltonians on curved spaces from the classical Zernike system

We consider the classical momentum- or velocity-dependent two-dimensional Hamiltonian given by where q i and p i are generic canonical variables, γ n are arbitrary coefficients, and . For N = 2, being both different from zero, this reduces to the classical Zernike system. We prove that always provides a superintegrable system (for any value of γ n and N) by obtaining the corresponding constants of the motion explicitly, which turn out to be of higher-order in the momenta. Such generic results are not only applied to the Euclidean plane, but also to the sphere and the hyperbolic plane. In the latter curved spaces, is expressed in geodesic polar coordinates showing that such a new superintegrable Hamiltonian can be regarded as a superposition of the isotropic 1 : 1 curved (Higgs) oscillator with even-order anharmonic curved oscillators plus another superposition of higher-order momentum-dependent potentials. Furthermore, the symmetry algebra determined by the constants of the motion is also studied, giving rise to a th-order polynomial algebra. As a byproduct, the Hamiltonian is interpreted as a family of superintegrable perturbations of the classical Zernike system. Finally, it is shown that (and so the Zernike system as well) is endowed with a Poisson -coalgebra symmetry which would allow for further possible generalizations that are also discussed.

On higher-dimensional superintegrable systems: a new family of classical and quantum Hamiltonian models

We introduce a family of n-dimensional Hamiltonian systems which, contain, as special reductions, several superintegrable systems as the Tremblay–Turbiner–Winternitz system, a generalized Kepler potential and the anisotropic harmonic oscillator with Rosochatius terms. We conjecture that there exist special values in the space of parameters, apart from those leading to known cases, for which this new Hamiltonian family is superintegrable.

Minimally superintegrable systems in flat three-dimensional space are also linearizable

It is shown that all minimally superintegrable Hamiltonian systems in a three-dimensional flat space derived in the work of Evans [Phys. Rev. A 41, 5666–5676 (1990)] possess hidden symmetries leading to their linearization.

Dynamical symmetry algebras of two superintegrable two-dimensional systems

A complete classification of 2D quadratically superintegrable systems with scalar potential on two-dimensional conformally flat spaces has been performed over the years and 58 models, divided into 12 equivalence classes, have been obtained. We will re-examine two pseudo-Hermitian quantum systems E 8 and E 10 from such a classification by a new approach based on extra sets of ladder operators. They correspond in fact to two of those equivalence classes. Those extra ladder operators are exploited to obtain the generating spectrum algebra and the dynamical symmetry one. We will relate the generators of the dynamical symmetry algebra to the Hamiltonian, thus demonstrating that the latter can be written in an algebraic form. We will also link them to the integrals of motion providing the superintegrability property. This demonstrates how the underlying dynamical symmetry algebra allows to write the integrals in terms of its generators and therefore explains the symmetries. Furthermore, we will exploit those algebraic constructions to generate extended sets of states and give the action of the ladder operators on them. We will present polynomials of the Hamiltonian and the integrals of motion that vanish on some of those states, then demonstrating that the sets of states not only contain eigenstates, but also generalized states which are beyond the well-known eigenstates of diagonalizable Hamiltonians and satisfy more complicated polynomial identities. Our approach provides a natural framework for such states.

Algebraic derivation of the energy eigenvalues for the quantum oscillator defined on the sphere and the hyperbolic plane

We give an algebraic derivation of eigenvalues of energy of a quantum harmonic oscillator on the surface of constant curvature, i.e., on the sphere or on the hyperbolic plane. We use the method proposed by Daskaloyannis [J. Math. Phys. 42, 1100–1119 (2001)] for fixing the energy eigenvalues of two-dimensional quadratically superintegrable systems by assuming that they are determined by the existence of a finite-dimensional representation of the polynomial algebra of motion integral operators. The tool for realizing representations is the deformed parafermionic oscillator. The eigenvalues of energy are calculated, and the result derived by us algebraically agrees with the known energy eigenvalues calculated by using classical analytical methods. This assertion, which is the main result of this article, is demonstrated by a detailed presentation. We also discuss the qualitative difference of the energy spectra on the sphere and on the hyperbolic plane.

More on Superintegrable Models on Spaces of Constant Curvature

A known general class of superintegrable systems on 2D spaces of constant curvature can be defined by potentials separating in (geodesic) polar coordinates. The radial parts of these potentials correspond either to an isotropic harmonic oscillator or a generalised Kepler potential. The angular components, on the contrary, are given implicitly by a transcendental, in general, equation. In the present note, devoted to the previously less studied models with the radial potential of the generalised Kepler type, a new two-parameter family of relevant angular potentials is constructed in terms of elementary functions. For an appropriate choice of parameters, the family reduces to an asymmetric spherical Higgs oscillator.

Polynomial algebras of superintegrable systems separating in Cartesian coordinates from higher order ladder operators

We introduce the general polynomial algebras characterizing a class of higher order superintegrable systems that separate in Cartesian coordinates. The construction relies on underlying polynomial Heisenberg algebras and their defining higher order ladder operators. One feature of these algebras is that they preserve by construction some aspects of the structure of the gl(n) Lie algebra. Among the classes of Hamiltonians arising in this framework are various deformations of harmonic oscillator and singular oscillator related to exceptional orthogonal polynomials and even Painlevé and higher order Painlevé analogs. As an explicit example, we investigate a new three-dimensional superintegrable system related to Hermite exceptional orthogonal polynomials of type III. Among the main results is the determination of the degeneracies of the model in terms of the finite-dimensional irreducible representations of the polynomial algebra.

Hydrogen atom in a magnetic field as an exactly solvable system without dynamical symmetries?

According to the generally accepted classification, there are supersymmetric superintegrable systems for which the Schrödinger equation is separable in more than one coordinate system and which admit an exact solution. A classic example of a superintegrable system is a hydrogen atom. It is believed that the converse is also true: an exactly solvable system has many integrals of motion corresponding to the separation constants of the Schrödinger equation in more than one coordinate system, which also means a high symmetry group of the system. It is shown that the existence of an exact solution for a hydrogen atom in an arbitrary strong uniform magnetic field apparently contradicts this classification.

Doubly Exotic Nth-Order Superintegrable Classical Systems Separating in Cartesian Coordinates

Superintegrable classical Hamiltonian systems in two-dimensional Euclidean space $E_2$ are explored. The study is restricted to Hamiltonians allowing separation of variables $V(x,y)=V_1(x)+V_2(y)$ in Cartesian coordinates. In particular, the Hamiltonian $\cal H$ admits a polynomial integral of order $N>2$. Only doubly exotic potentials are considered. These are potentials where none of their separated parts obey any linear ordinary differential equation. An improved procedure to calculate these higher-order superintegrable systems is described in detail. The two basic building blocks of the formalism are non-linear compatibility conditions and the algebra of the integrals of motion. The case $N=5$, where doubly exotic confining potentials appear for the first time, is completely solved to illustrate the present approach. The general case $N>2$ and a formulation of inverse problem in Superintegrability are briefly discussed as well.

On integrable systems outside Nijenhuis and Haantjes geometry

We study non-invariant Killing tensors with non-zero Nijenhuis torsion in the three-dimensional Euclidean space. Generalizing the corresponding integrable systems we construct two new families of superintegrable systems in n-dimensional Euclidean space.

Heisenberg supermagnetic hierarchy with the quadratic and cubic constraints

This paper is concerned with the construction of the Heisenberg supermagnetic hierarchy with quadratic constraints S2=S,S2=3S−2I and the cubic constraint S3=S. Firstly, we establish an algebraic model and investigate an algorithm to compute recursion operators of Lax representation for generalized Heisenberg supermagnetic models with quadratic constraints, which make the process easier and more automatic. Furthermore, for the cubic constraint S3=S, an approach has been proposed to construct the higher-order Heisenberg supermagnetic models. As an application, we construct the third-order Heisenberg supermagnetic models on the superalgebra ospu(1,1/1) with constraint S3=S. Meanwhile, in terms of the gauge transformation, we derive the gauge equivalent counterpart, i.e., supersymmetric nonlinear Schrödinger equation. In addition, the Hamiltonian structure of the super and fermionic integrable systems has been discussed.

Multi-Hamiltonian structure of the epidemics model accounting for vaccinations and a suitable test for the accuracy of its numerical solvers

We derive a generalized Hamiltonian formalism for a modified susceptible–infectious–recovered/removed (SIR) epidemic model taking into account the population V of vaccinated persons. The resulting SIRV model is shown to admit three possible functionally independent Hamiltonians and hence three associated Poisson structures. The reduced case of vanishing vaccinated sector shows a complete correspondence with the known Poisson structures of the SIR model. The SIRV model is shown to be expressible as an almost Nambu system, except for a scale factor function breaking the divergenceless property. In the autonomous case with time-independent stationary ratios k and b, the SIRV model is shown to be a maximally super-integrable system. For this case we test the accuracy of numerical schemes that are suited to solve the stiff set of SIRV differential equations.

A family of fourth-order superintegrable systems with rational potentials related to Painlevé VI

We discuss a family of Hamiltonians given by particular rational extensions of the singular oscillator in two-dimensions. The wave functions of these Hamiltonians can be expressed in terms of products of Laguerre and exceptional Jacobi polynomials. We show that these systems are superintegrable and admit an integral of motion that is of fourth-order. As such systems have been classified, we see that these potentials satisfy a non-linear equation related to Painlevé VI. We begin by demonstrating the process with the simpler example of rational extensions of the harmonic oscillator and use the classification of third-order superintegrable systems to connect these families with the known solutions of Painlevé IV associated with Hermite polynomials.

On some algebraic formulations within universal enveloping algebras related to superintegrability

We report on some recent purely algebraic approaches to superintegrable systems from the perspective of subspaces of commuting polynomials in the enveloping algebras of Lie algebras that generate quadratic (and eventually higher-order) algebras. In this context, two algebraic formulations are possible; a first one strongly dependent on representation theory, as well as a second formal approach that focuses on the explicit construction within commutants of algebraic integrals for appropriate algebraic Hamiltonians defined in terms of suitable subalgebras. The potential use in this context of the notion of virtual copies of Lie algebras is briefly commented.

Symmetries of Schrödinger–Pauli equations for charged particles and quasirelativistic Schrödinger equations

Lie symmetries of Schrödinger–Pauli equations for charged particles and quasirelativistic Schrödinger equations are classified. In particular a new superintegrable system with spin–orbit coupling is discovered.

Kähler geometry for su(1,N|M) superconformal mechanics

We suggest the $su(1,N|M)$ superconformal mechanics formulated in terms of phase superspace given by the noncompact analogue of complex projective superspace. We parametrized this phase space by the specific coordinates allowing us to interpret it as a higher-dimensional superanalogue of the Lobachevsky plane parametrized by lower half-plane (Klein model). Then we introduced the canonical coordinates corresponding to the known separation of the ``radial'' and ``angular'' parts of (super)conformal mechanics. Relating the ``angular'' coordinates with action-angle variables, we demonstrated that the proposed scheme allows us to construct the $su(1,N|M)$ supeconformal extensions of wide class of superintegrable systems. We also proposed the superintegrable oscillator- and Coulomb-like systems with a $su(1,N|M)$ dynamical superalgebra and found that oscillatorlike systems admit deformed $\mathcal{N}=2M$ Poincar\'e supersymmetry, in contrast with Coulomb-like ones.

Superintegrable Bertrand Magnetic Geodesic Flows

The problem of description of superintegrable systems (i.e., systems with closed trajectories in a certain domain) in the class of rotationally symmetric natural mechanical systems goes back to Bertrand and Darboux. We describe all superintegrable (in a domain of slow motions) systems in the class of rotationally symmetric magnetic geodesic flows. We show that all sufficiently slow motions in a central magnetic field on a two-dimensional manifold of revolution are periodic if and only if the metric has a constant scalar curvature and the magnetic field is homogeneous, i.e. proportional to the area form.

Rational integrals of 2-dimensional geodesic flows: New examples

This paper is devoted to searching for Riemannian metrics on 2-surfaces whose geodesic flows admit a rational in momenta first integral with a linear numerator and denominator. The explicit examples of metrics and such integrals are constructed. Few superintegrable systems are found having both a polynomial and a rational integrals which are functionally independent of the Hamiltonian.

Superintegrability of separable systems with magnetic field: the cylindrical case

We present a general method simplifying the search for additional integrals of motion of three dimensional systems with magnetic fields. The method is suitable for systems possessing at least one conserved canonical momentum in a suitable coordinates system. It reduces the problem either to consideration of lower dimensional systems or of particular constrained forms of the hypothetical integral. In particular, it is applicable to all separable systems in the Euclidean space since they are known to possess at least one cyclic coordinates when magnetic field is present. Next, we focus on systems which separate in the cylindrical coordinates. Using our method, we are able to classify all superintegrable systems of this kind under the assumption that all considered integrals are at most second order in the momenta. In addition to already known systems, several new minimally superintegrable systems are found and we show that no quadratically maximally superintegrable ones can exist. We also construct some examples of systems with higher order integrals.