Second order superintegrable systems in conformally flat spaces. II. The classical two-dimensional Stäckel transform

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

doi:10.1063/1.1894985

Cited by 98 publications

(181 citation statements)

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“…(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%

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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%

“…If λ is the metric of a space that admits a nondegenerate superintegrable system, then it is always possible to choose coordinates x, y such that λ 12 = 0 [41]. In [25] we prove the following basic result. where either…”

Section: The Stäckel Transform For Two-dimensional Systemsmentioning

confidence: 99%

Jacobi, Ellipsoidal Coordinates and Superintegrable Systems

Kalnins¹,

Kress²,

Miller³

2005

JNMP

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“…This suggests that to classify all such superintegrable systems we can restrict attention to these two constant curvature spaces, and then obtain all other cases via Stäckel transforms. We are making considerable progress on the classification theory [21], though the problem is complicated.…”

Section: Discussionmentioning

confidence: 99%

“…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%

“…The key difference with our first example is, as we shall show later, that the 3-parameter Kepler-Coulomb potential is degenerate and it cannot be extended to a 4-parameter potential. In [20,21] there are examples of superintegrable systems on the 3-sphere that admit a quadratic algebra structure. A more general set of examples arises from a space with metric…”

Section: Introduction and Examplesmentioning

confidence: 99%

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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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