The problem of finding superintegrable Hamiltonians and their integrals of motion can be reduced to solving a series of compatibility equations that result from the overdetermination of the commutator or Poisson bracket relations. The computation of the compatibility equations requires a general formula for the coefficients, which in turn must depend on the potential to be solved for. This is in general a nonlinear problem and quite difficult. Thus, research has focused on dividing the classes of potential into standard and exotic ones so that a number of parameters may be set to zero and the coefficients may be obtained in a simpler setting. We have developed a new method in both the classical and quantum setting which readily yields a formula for the coefficients of the invariant without recourse to this division in the case of Cartesian-separable Hamiltonians. Even though they allow separation of variables as they in general involve potential in terms of higher transcendental and beyond hypergeometric for their wavefunctions, they are quite non-trivial models. The expressions we obtain are in general non-polynomial in the momenta whose fractional terms can be arbitrarily set to zero. These conditions are equivalent to the compatibility equations, but the only unknowns in addition to the potential are constant parameters. We also give the fourth-order standard potentials, and conjectures about general families.
We consider superintegrability in classical mechanics in the presence of magnetic fields. We focus on three-dimensional systems which are separable in Cartesian coordinates. We construct all possible minimally and maximally superintegrable systems in this class with additional integrals quadratic in the momenta. Together with the results of our previous paper [J. Phys. A: Math. Theor. 50 (2017), 245202, 24 pages], where one of the additional integrals was by assumption linear, we conclude the classification of three-dimensional quadratically minimally and maximally superintegrable systems sepa…
A study is presented of two-dimensional superintegrable systems separating in Cartesian coordinates and allowing an integral of motion that is a fourth order polynomial in the momenta. All quantum mechanical potentials that do not satisfy any linear differential equation are found. They do however satisfy nonlinear ODEs. We show that these equations always have the Painlevé property and integrate them in terms of known Painlevé transcendents or elliptic functions.
Heisenberg-type higher order symmetries are studied for both classical and quantum mechanical systems separable in cartesian coordinates. A few particular cases of this type of superintegrable systems were already considered in the literature, but here they are characterized in full generality together with their integrability properties. Some of these systems are defined only in a region of $\mathbb R^n$, and in general they do not include bounded solutions. The quantum symmetries and potentials are shown to reduce to their superintegrable classical analogs in the $\hbar \to0$ limit.
We continue the study of superintegrable systems of Thompson's type separable in Cartesian coordinates. An additional integral of motion for these systems is the polynomial in momenta of N-th order which is a linear function of angle variables and the polynomial in action variables. Existence of such superintegrable systems is naturally related to the famous Chebyshev theorem on binomial differentials.
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 $\mathcal H$ 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 th…
We develop new constructions of 2D classical and quantum superintegrable Hamiltonians allowing separation of variables in Cartesian coordinates. In classical mechanics we start from two functions on a one-dimensional phase space, a natural Hamiltonian $H$ and a polynomial of order $N$ in the momentum $p.$ We assume that their Poisson commutator $\{H,K\}$ vanishes, is a constant, a constant times $H$, or a constant times $K$. In the quantum case $H$ and $K$ are operators and their Lie commutator has one of the above properties. We use two copies of such $(H,K)$ pairs to generate two-dimensional…
A study is presented of superintegrable quantum systems in two-dimensional Euclidean space $E_2$ allowing the separation of variables in Cartesian coordinates. In addition to the Hamiltonian $H$ and the second order integral of motion $X$, responsible for the separation of variables, they allow a third integral that is a polynomial of order $N\, (N\geq3)$ in the components $p_1, p_2$ of the linear momentum. We focus on doubly exotic potentials, i.e. potentials $V(x, y) = V_1(x) + V_2(y)$ where neither $V_1(x)$ nor $V_2(y)$ satisfy any linear ordinary differential equation. We present two new i…
The conditions for superintegrable systems in two-dimensional Euclidean space admitting separation of variables in an orthogonal coordinate system and a functionally independent third-order integral are studied. It is shown that only systems that separate in subgroup type coordinates, Cartesian or polar, admit potentials that can be described in terms of nonlinear special functions. Systems separating in parabolic or elliptic coordinated are shown to have potentials with only non-movable singularities.
We present in this article all Hamiltonian systems in E(2) that are separable in cartesian coordinates and that admit a third-order integral, both in quantum and in classical mechanics. Many of these superintegrable systems are new, and it is seen that there exists a relation between quantum superintegrable potentials, invariant solutions of the Korteweg-De Vries equation and the Painlevé transcendents.
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.
We present polynomial Poisson algebras for the 8 classical potentials in two-dimensional Euclidian space that separate in cartesian coordinates and allow a third order integral of motion. Some of the classical superintegrale potentials do not coincide with quantum ones, but are their singular limits. We show that all bounded trajectories in these potentials are periodic.
In this paper, we investigate superintegrable systems which separate in parabolic coordinates and admit a third-order integral of motion. We give the corresponding determining equations and show that all such systems are multi-separable and so admit two second-order integrals. The third-order integral is their Lie or Poisson commutator. We discuss how this situation is different from the Cartesian and polar cases where new potentials were discovered which are not multi-separable and which are expressed in terms of Painlevé transcendents or elliptic functions.
We extend the construction of 2D superintegrable Hamiltonians with separation of variables in spherical coordinates using combinations of shift, ladder, and supercharge operators to models involving rational extensions of the two-parameter Lissajous systems on the sphere. These new families of superintegrable systems with integrals of arbitrary order are connected with Jacobi exceptional orthogonal polynomials (EOP) of type I (or II) and supersymmetric quantum mechanics (SUSYQM). Moreover, we present an algebraic derivation of the degenerate energy spectrum for the one- and two-parameter Lissa…
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 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 quantum reducible and irreducible rational potentials separable in Cartesian coordinates in E2. We also discuss these potentials from the point of vi…
We consider a two dimensional quantum Hamiltonian separable in Cartesian coordinates and allowing a fifth-order integral of motion. We impose the superintegrablity condition and find all doubly exotic superintegrable potentials (i.e potentials V (x, y) = V_1(x)+V_2(y) where neither V_1(x) nor V_2(y) satisfy a linear ODE) allowing the existence of such an integral. All of these potentials are found to have the Painlevé property. Most of them are expressed in terms of known Painlevé transcendents or elliptic functions but some may represent new higher order Painlevé transcendents.
Superintegrable systems are classical and quantum Hamiltonian systems which enjoy much symmetry and structure that permit their solubility via analytic and even, algebraic means. They include such well-known and important models as the Kepler potential, Calogero-Moser model, and harmonic oscillator, as well as its integrable perturbations, for example, the Smorodinsky-Winternitz (SW) potential. Normally, the problem of classification of superintegrable systems is approached by considering associated algebraic, or geometric structures. To this end we invoke the invariant theory of Killing tenso…
The goal of this thesis is the search for integrable and superintegrable systems with magnetic field. We formulate the quantum mechanical determining equations for second order integrals of motion in the cylindrical coordinates and we find all quadratically integrable systems of the cylindrical type. Among them we search for systems admitting additional integrals of motion. We find all systems with an additional first order integral both in classical and quantum mechanics. It turns out that all these systems have already been known and no other exist. We also find all systems with an additiona…
In recent years, progress toward the classification of superintegrable systems with higher order integrals of motion has been made. In particular, a complete classification of all exotic potentials with a third or a fourth order integrals, and allowing separation of variables in Cartesian coordinates. All doubly exotic potentials with a fifth order integral have also been completely classified. It has been demonstrated how the Chazy class of third order differential equations plays an important role in solving determining equations. Moreover, taking advantage of various operator algebras defin…
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 $N>2$. 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…
We discuss the properties of superintegrable Hamiltonian systems, in particular those that admit separation of variables in cartesian coordinates. We show that the superintegrability of such potentials is equivalent to the isochronicity of the separated potentials. We use this fact to get a new insight into an old question about the relation between quantum and classical harmonic behavior.
This article deals with a quantum-mechanical system which generalizes the ordinary isotropic harmonic oscillator system. We give the coefficients connecting the polar and Cartesian bases for D=2 and the coefficients connecting the Cartesian and cylindrical bases as well as the cylindrical and spherical bases for D=3. These interbasis expansion coefficients are found to be analytic continuations to real values of their arguments of the Clebsch-Gordan coefficients for the group SU(2). For D=2, the superintegrable character for the generalized oscillator system is investigated from the points of …
