Maximally superintegrable Smorodinsky-Winternitz systems on the N-dimensional sphere and hyperbolic spaces
Francisco J. Herranz, Angel Ballesteros, Mariano Santander, Teresa Sanz-Gil
备注：
19 pages; two figures. Published in "Superintegrability in Classical and Quantum Systems", edited by P.Tempesta, P.Winternitz, J.Harnad, W.Miller Jr., G.Pogosyan and M.A.Rodriguez, CRM Proceedings & Lecture Notes, vol.37, American Mathematical Society, 2004
发表信息：
Superintegrability in Classical and Quantum Systems. P. Tempesta et al (eds). CRM Proceedings & Lecture Notes, vol. 37, AMS (2004) 75-89
The classical Smorodinsky-Winternitz systems on the ND sphere, Euclidean and hyperbolic spaces S^N, E^N and H^N are simultaneously approached starting from the Lie algebras so_k(N+1), which include a parametric dependence on the curvature k. General expressions for the Hamiltonian and its integrals of motion are given in terms of intrinsic geodesic coordinate systems. Each Lie algebra generator gives rise to an integral of motion, so that a set of N(N+1)/2 integrals is obtained. Furthermore, 2N-1 functionally independent ones are identified which, in turn, shows that the well known maximal superintegrability of the Smorodinsky-Winternitz system on E^N is preserved when curvature arises. On both S^N and H^N, the resulting system can be interpreted as a superposition of an "actual" oscillator and N "ideal" oscillators (for the sphere, these are alike the actual ones), which can also be understood as N "centrifugal terms"; this is the form seen in the Euclidean limiting case.
We employ joint invariants of Killing two-tensors defined in the Euclidean plane to characterize the Smorodinsky-Winternitz potential and explain the geometric meaning of its arbitrary parameters. In addition, we verify for which values of the arbitrary parameter $k$ the Tremblay-Turbiner-Winternitz potential is multi-separable.
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 $\mathbb{C}^2$- 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 fiel…
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 pote…
There exist a relation between the Klein-Gordon and the Dirac equations with scalar and vector potentials of equal magnitude (SVPEM) and the Schrodinger equation. We obtain the relativistic energy spectrum for the four Smorodinsky-Winternitz systems from the quasi-Hamiltonian and the quadratic algebras obtained by Daskaloyannis in the non-relativistic context. We point out how results obtained in context of quantum superintegrable systems and their polynomial algebras may be applied to the quantum relativistic case. We also present the symmetry algebra of the Dirac equation for these four syst…
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 ${\cal SW}(N)$ of the system, which it is a higher-rank quadratic one containing the recently discovered Racah algebra ${\cal R}(N)$ as subalgebra. The substructures of distinct quadratic ${\cal Q}(3)$ algebras and their related Casimirs are also studied. In this way, fro…
The 2D Smorodinsky--Winternitz II system and the Laguerre--Heun algebra
Centre de recherches mathématiques, Université de Montréal（蒙特利尔大学数学研究中心）；Département de physique, Université de Montréal（蒙特利尔大学物理系）；Euler International Mathematical Institute（欧拉国际数学研究所）
本文识别了二维 Smorodinsky--Winternitz II 系统的二次对称代数与 Laguerre 型合流 Heun 代数，通过笛卡尔和抛物线坐标下的分离性以及互补算子的代数关系，给出了该系统的直接超可积实现。
It is known that the fairly (most?) general class of 2D superintegrable systems defined on 2D spaces of constant curvature and separating in (geodesic) polar coordinates is specified by two types of radial potentials (oscillator or (generalized) Kepler ones) and by corresponding families of angular potentials. Unlike the radial potentials the angular ones are given implicitly (up to a function) by, in general, transcendental equation. In the present paper new two-parameter families of angular potentials are constructed in terms of elementary functions. It is shown that for an appropriate choic…
The higher-order superintegrability of separable potentials is studied. It is proved that these potentials possess (in addition to the two quadratic integrals) a third integral of higher-order in the momenta that can be obtained as the product of powers of two particular rather simple complex functions. Some systems related with the harmonic oscillator, as the generalized SW system and the TTW system, were studied in previous papers; now a similar analysis is presented for superintegrable systems related with the Kepler problem. In this way, a new proof of the superintegrability of the Post-Wi…
Superintegrable Hamiltonian systems in a two-dimensional Euclidean space are considered. We present all real standard potentials that allow separation of variables in polar coordinates and admit an independent fourth-order integral of motion. The general form of the potentials satisfies a linear ODE. In the classical case, the standard potentials coincide with the Tremblay-Turbiner-Winternitz (TTW) or Post-Winternitz (PW) models. In the quantum case new superintegrable systems are obtained, in addition to the TTW and PW ones. Their classical limit is free motion.
Several examples of classical superintegrable systems in two-dimensional spac are shown to possess hidden symmetries leading to their linearization. They are those determined 50 years ago in [Phys. Lett. 13, 354 (1965)], and the more recent Tremblay-Turbiner-Winternitz system [J. Phys. A: Math. Theor. 42, 242001 (2009)]. We conjecture that all classical superintegrable systems in two-dimensional space have hidden symmetries that make them linearizable.
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.
The existence of quasi-bi-Hamiltonian structures for a two-dimensional superintegrable $(k_1,k_2,k_3)$-dependent Kepler-related problem is studied. We make use of an approach that is related with the existence of some complex functions which satisfy interesting Poisson bracket relations and that was previously applied to the standard Kepler problem as well as to some particular superintegrable systems as the Smorodinsky-Winternitz (SW) system, the Tremblay-Turbiner-Winternitz (TTW) and Post-Winternitz (PW) systems. We prove that these complex functions are important for two reasons: first, the…
Construction and classification of 2D superintegrable systems (i.e. systems admitting, in addition to two global integrals of motion guaranteeing the Liouville integrability, the third global and independent one) defined on 2D spaces of constant curvature and separable in the so called geodesic polar coordinates are presented. The method proposed is applicable to any value of curvature including the case of Euclidean plane, the 2-sphere and the hyperbolic plane. The mathematic used is essentially "physical", in particular it refers to the very elegant technique of action-angle variables and pe…
The factorization technique for superintegrable Hamiltonian systems is revisited and applied in order to obtain additional (higher-order) constants of the motion. In particular, the factorization approach to the classical anisotropic oscillator on the Euclidean plane is reviewed, and new classical (super)integrable anisotropic oscillators on the sphere are constructed. The Tremblay-Turbiner-Winternitz system on the Euclidean plane is also studied from this viewpoint.
We generalize the idea of "extension of Hamiltonian systems" -- developed in a series of previous articles -- which allows the explicit construction of Hamiltonian systems with additional non-trivial polynomial first integrals of arbitrarily high degree, as well as the determination of new superintegrable systems from old ones. The present generalization, that we call "modified extension of Hamiltonian systems", produces the third independent first integral for the (complete) Tremblay-Turbiner-Winternitz (TTW) system, as well as for the caged anisotropic oscillator in dimension two.
We describe a method for determining a complete set of integrals for a classical Hamiltonian that separates in orthogonal subgroup coordinates. As examples, we use it to determine complete sets of integrals, polynomial in the momenta, for some families of generalized oscillator and Kepler-Coulomb systems, hence demonstrating their superintegrability. The latter generalizes recent results of Verrier and Evans, and Rodriguez, Tempesta and Winternitz. Another example is given of a superintegrable system on a non-conformally flat space.
Nineteen classical superintegrable systems in two-dimensional non-Euclidean spaces are shown to possess hidden symmetries leading to their linearization. They are the two Perlick systems [A. Ballesteros, A. Enciso, F.J. Herranz and O. Ragnisco, Class. Quantum Grav. 25, 165005 (2008)], the Taub-NUT system [A. Ballesteros, A. Enciso, F.J. Herranz, O. Ragnisco, and D. Riglioni, SIGMA 7, 048 (2011)], and all the seventeen superintegrable systems for the four types of Darboux spaces as determined in [E.G. Kalnins, J.M. Kress, W. Miller, P. Winternitz, J. Math. Phys. 44, 5811--5848 (2003)].
We prove the integrability and superintegrability of a family of natural Hamiltonians which includes and generalises those studied in some literature, originally defined on the 2D Minkowski space. Some of the new Hamiltonians are a perfect analogy of the well-known superintegrable system on the Euclidean plane proposed by Tremblay-Turbiner-Winternitz and they are defined on Minkowski space, as well as on all other 2D manifolds of constant curvature, Riemannian or pseudo-Riemannian. We show also how the application of the coupling-constant-metamorphosis technique allows us to obtain new superin…
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…
We revise a method by Kalnins, Kress and Miller (2010) for constructing a canonical form for symmetry operators of arbitrary order for the Schrödinger eigenvalue equation $HΨ\equiv (Δ_2 +V)Ψ=EΨ$ on any 2D Riemannian manifold, real or complex, that admits a separation of variables in some orthogonal coordinate system. Most of this paper is devoted to describing the method. Details will be provided elsewhere. As examples we revisit the Tremblay and Winternitz derivation of the Painlevé VI potential for a 3rd order superintegrable flat space system that separates in polar coordinates and we show …
Superintegrable systems in two- and three-dimensional spaces of constant curvature have been extensively studied. From these, superintegrable systems in conformally flat spaces can be constructed by Staeckel transform. In this paper a method developed to establish the superintegrability of the Tremblay-Turbiner-Winternitz system in two dimensions is extended to higher dimensions and a superintegrable system on a non-conformally-flat four-dimensional space is found. In doing so, curvature corrections to the corresponding classical potential are found to be necessary. It is found that some subal…
Recently the authors and J.M. Kress presented a special function recurrence relation method to prove quantum superintegrability of an integrable 2D system that included explicit constructions of higher order symmetries and the structure relations for the closed algebra generated by these symmetries. We applied the method to 5 families of systems, each depending on a rational parameter k, including most notably the caged anisotropic oscillator, the Tremblay, Turbiner and Winternitz system and a deformed Kepler-Coulomb system. Here we work out the analogs of these constructions for all of the as…
We revise a method by Kalnins, Kress and Miller (2010) for constructing a canonical form for symmetry operators of arbitrary order for the Schrödinger eigenvalue equation $HΨ\equiv (Δ_2 +V)Ψ=EΨ$ on any 2D Riemannian manifold, real or complex, that admits a separation of variables in some orthogonal coordinate system. We apply the method, as an example, to revisit the Tremblay and Winternitz (2010) derivation of the Painlevé VI potential for a 3rd order superintegrable flat space system that separates in polar coordinates and, as new results, we give a listing of the possible potentials on the …
The quantum Kepler-Coulomb system in 3 dimensions is well known to be 2nd order superintegrable, with a symmetry algebra that closes polynomially under commutators. This polynomial closure is also typical for 2nd order superintegrable systems in 2D and for 2nd order systems in 3D with nondegenerate (4-parameter) potentials. However the degenerate 3-parameter potential for the 3D Kepler-Coulomb system (also 2nd order superintegrable) is an exception, as its symmetry algebra doesn't close polynomially. The 3D 4-parameter potential for the extended Kepler-Coulomb system is not even 2nd order supe…
The classical Kepler-Coulomb system in 3 dimensions is well known to be 2nd order superintegrable, with a symmetry algebra that closes polynomially under Poisson brackets. This polynomial closure is typical for 2nd order superintegrable systems in 2D and for 2nd order systems in 3D with nondegenerate (4-parameter) potentials. However the degenerate 3-parameter potential for the 3D extended Kepler-Coulomb system (also 2nd order superintegrable) is an exception, as its quadratic symmetry algebra doesn't close polynomially. The 3D 4-parameter potential for the extended Kepler-Coulomb system is no…
An Exactly-Solvable (ES) potential on the sphere $S^n$ is reviewed and the related Quasi-Exactly-Solvable (QES) potential is found and studied. Mapping the sphere to a simplex it is found that the metric (of constant curvature) is in polynomial form, and both the ES and the QES potentials are rational functions. Their hidden algebra is $gl_n$ in a finite-dimensional representation realized by first order differential operators acting on $RP^n$. It is shown that variables in the Schrödinger eigenvalue equation can be separated in spherical coordinates and a number of the integrals of the second…
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 $2$D can be obtained by means of suitable limits and contractions. A higher rank generalization 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 naturall…
Oscillator and Coulomb systems on N-dimensional spaces of constant curvature can be generalized by replacing their angular degrees of freedom with a compact integrable (N-1)-dimensional system. We present the action-angle formulation of such models in terms of theradial degree of freedom and the action-angle variables of the angular subsystem. As an example, we construct the spherical and pseudospherical generalization of the two-dimensional superintegrable models introduced by Tremblay, Turbiner and Winternitz and by Post and Winternitz. We demonstrate the superintegrability of these systems …
Classical (maximal) superintegrable systems in $n$ dimensions are Hamiltonian systems with $2n-1$ independent constants of the motion, globally defined, the maximum number possible. They are very special because they can be solved algebraically. In this paper we show explicitly, mostly through examples of 2nd order superintegrable systems in 2 dimensions, how the trajectories can be determined in detail using rather elementary algebraic, geometric and analytic methods applied to the closed quadratic algebra of symmetries of the system. We treat a family of 2nd order degenerate systems: oscilla…
The 5D Kepler system possesses many interesting properties. This system is superintegrable and also with a $su(2)$ nonAbelian monopole interaction (Yang-Coulomb monopole). This system is also related to a 8D isotropic harmonic oscillator by a Hurwitz transformation. We introduce a new superintegrable Hamiltonian that consists in a 5D Kepler system with new terms of Smorodinsky-Winternitz type. We obtain the integrals of motion of this systems. They generate a quadratic algebra with structure constants involving the Casimir operator of a $so(4)$ Lie algebra. We also show that this system remain…
In a recent FTC by Tremblay {\sl et al} (2009 {\sl J. Phys. A: Math. Theor.} {\bf 42} 205206), it has been conjectured that for any integer value of $k$, some novel exactly solvable and integrable quantum Hamiltonian $H_k$ on a plane is superintegrable and that the additional integral of motion is a $2k$th-order differential operator $Y_{2k}$. Here we demonstrate the conjecture for the infinite family of Hamiltonians $H_k$ with odd $k \ge 3$, whose first member corresponds to the three-body Calogero-Marchioro-Wolfes model after elimination of the centre-of-mass motion. Our approach is based on…
Construction of superintegrable systems based on Lie algebras have been introduced over the years. However, these approaches depend on explicit realisations, for instance as a differential operators, of the underlying Lie algebra. This is also the case for the construction of their related symmetry algebra which take usually the form of a finitely generated quadratic algebra. These algebras often display structure constants which depend on the central elements and in particular on the Hamiltonian. In this paper, we develop a new approach reexamining the case of the generic superintegrable syst…
Families of three-body Hamiltonian systems in one dimension have been recently proved to be maximally superintegrable by interpreting them as one-body systems in the three-dimensional Euclidean space, examples are the Calogero, Wolfes and Tramblay Turbiner Winternitz systems. For some of these systems, we show in a new way how the superintegrability is associated with their dihedral symmetry in the three-dimensional space, the order of the dihedral symmetries being associated with the degree of the polynomial in the momenta first integrals. As a generalization, we introduce the analysis of int…
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 (S-W) system and the Harmonic Oscillator with ratio of frequencies 1:1:2 and additional nonlinear terms on the 3-dimensional sphere $S^3$ ($\kp>0)$ and on the hyperbolic space $H^3$ ($\kp<0$). In the second part we present a study first of the Kepler problem and then …
