Second-order superintegrable systems in conformally flat spaces. I. Two-dimensional classical structure theory

Kalnins, E. G.; Kress, Jonathan M.; Miller, Willard

doi:10.1063/1.1897183

Cited by 111 publications

(207 citation statements)

References 37 publications

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“…This is a major complication. In [27] we overcome this problem by proving a 5 =⇒ 6 Theorem, that is, five functionally independent second-order symmetries for a nondegenerate superintegrable three-dimensional system imply six linearly independent second-order symmetries. Then we demonstrate that for three-dimensional conformally flat superintegrable systems with nondegenerate potential the maximum possible dimensions of the spaces of second-, third-, fourth-and sixth-order symmetries are six, four, 26 and 56, respectively, and these dimensions are achieved.…”

Section: Conclusion and Further Resultsmentioning

confidence: 99%

“…(In [26,25] the details of the proofs are given and the results are extended to systems with degenerate potentials.) We have shown that all these systems are Stäckel equivalent to superintegrable systems on spaces of constant curvature, the potentials of which have already been classified in detail [36,30,29].…”

Section: Conclusion and Further Resultsmentioning

confidence: 99%

“…The passage from the three-dimensional conformally flat classical superintegrable systems to quantum superintegrable systems is still straightforward, but requires modifying the quantum potential by an additive term proportional to the scalar curvature [28]. Work is in progress to determine all three-dimensional superintegrable systems.…”

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confidence: 99%

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Jacobi, Ellipsoidal Coordinates and Superintegrable Systems

Kalnins¹,

Kress²,

Miller³

2005

JNMP

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We describe Jacobi's method for integrating the Hamilton-Jacobi equation and his discovery of elliptic coordinates, the generic separable coordinate systems for real and complex constant curvature spaces. This work was an essential precursor for the modern theory of second-order superintegrable systems to which we then turn. A Schrödinger operator with potential on a Riemannian space is second-order superintegrable if there are 2n − 1 (classically) functionally independent second-order symmetry operators. (The 2n − 1 is the maximum possible number of such symmetries.) These systems are of considerable interest in the theory of special functions because they are multiseparable, i.e., variables separate in several coordinate sets and are explicitly solvable in terms of special functions. The interrelationships between separable solutions provides much additional information about the systems. We give an example of a superintegrable system and then present very recent results exhibiting the general structure of superintegrable systems in all real or complex two-dimensional spaces and three-dimensional conformally flat spaces and a complete list of such spaces and potentials in two dimensions.

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Section: Conclusion and Further Resultsmentioning

confidence: 99%

Section: Conclusion and Further Resultsmentioning

confidence: 99%

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Jacobi, Ellipsoidal Coordinates and Superintegrable Systems

Kalnins¹,

Kress²,

Miller³

2005

JNMP

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“…Two dimensional second order superintegrable systems have been studied and classified by the author and his collaborators in a recent series of papers [18,19,20,21]. Here we concentrate on three dimensional (3D) systems where new complications arise.…”

Section: Introduction and Examplesmentioning

confidence: 99%

Second Order Superintegrable Systems in Three Dimensions

Miller¹

2005

SIGMA

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Abstract. A classical (or quantum) superintegrable system on an n-dimensional Riemannian manifold is an integrable Hamiltonian system with potential that admits 2n − 1 functionally independent constants of the motion that are polynomial in the momenta, the maximum number possible. If these constants of the motion are all quadratic, the system is second order superintegrable. Such systems have remarkable properties. Typical properties are that 1) they are integrable in multiple ways and comparison of ways of integration leads to new facts about the systems, 2) they are multiseparable, 3) the second order symmetries generate a closed quadratic algebra and in the quantum case the representation theory of the quadratic algebra yields important facts about the spectral resolution of the Schrödinger operator and the other symmetry operators, and 4) there are deep connections with expansion formulas relating classes of special functions and with the theory of Exact and Quasi-exactly Solvable systems. For n = 2 the author, E.G. Kalnins and J. Kress, have worked out the structure of these systems and classified all of the possible spaces and potentials. Here I discuss our recent work and announce new results for the much more difficult case n = 3. We consider classical superintegrable systems with nondegenerate potentials in three dimensions and on a conformally flat real or complex space. We show that there exists a standard structure for such systems, based on the algebra of 3 × 3 symmetric matrices, and that the quadratic algebra always closes at order 6. We describe the Stäckel transformation, an invertible conformal mapping between superintegrable structures on distinct spaces, and give evidence indicating that all our superintegrable systems are Stäckel transforms of systems on complex Euclidean space or the complex 3-sphere. We also indicate how to extend the classical 2D and 3D superintegrability theory to include the operator (quantum) case.

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“…In the last few years we have witnessed [3,4,8,5] considerable interest in Hamiltonian systems that admit a maximal number of 2n − 1 functionally independent integrals of motion. Their trajectories are completely determined as the intersection of surfaces of constant value of these integrals of motion.…”

Section: Introductionmentioning

confidence: 99%

Triangular Newton Equations with Maximal Number of Integrals of Motion

Persson

Rauch-Wojciechowski

2005

JNMP

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We study two-dimensional triangular systems of Newton equations (acceleration = velocity-independent force) admitting three functionally independent quadratic integrals of motion. The main idea is to exploit the fact that the first component M 1 (q 1 ) of a triangular force depends on one variable only. By using the existence of extra integrals of motion we reduce the problem to solving a simultaneous system of three linear ordinary differential equations with nonconstant coefficients for M 1 (q 1 ). With the help of computer algebra we have found and solved these ordinary differential equations in all cases. A complete list of superintegrable triangular equations in two dimensions is been given. Most of these equations were not known before.

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Second-order superintegrable systems in conformally flat spaces. I. Two-dimensional classical structure theory

Cited by 111 publications

References 37 publications

Jacobi, Ellipsoidal Coordinates and Superintegrable Systems

Jacobi, Ellipsoidal Coordinates and Superintegrable Systems

Second Order Superintegrable Systems in Three Dimensions

Triangular Newton Equations with Maximal Number of Integrals of Motion

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