An algebraic geometric classification of superintegrable systems in the Euclidean plane

Kress, Jonathan M.; Schöbel, Konrad

doi:10.1016/j.jpaa.2018.07.005

Cited by 15 publications

(44 citation statements)

References 41 publications

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“…While the introduction given here is self-contained, the reader might like to consult, in addition, a more detailed review of the topic, such as [KKM18]. For the algebraic geometric approach to second-order (properly) superintegrable systems see [Cap14,KKM07a,KKM07b] and particularly [KS18,KSV19].…”

Section: Introductionmentioning

confidence: 99%

“…We focus our discussion on second-order superintegrable systems whose potential satisfies the Wilczynski equation (introduced in [KSV19])…”

Section: Introductionmentioning

confidence: 99%

“…where T k ij is a tensor symmetric and tracefree in the first two indices. All systems in the known classification in dimensions 2 and 3 are of this kind [KKM05b, KKM05a, KKM05c, KKM06] and any irreducible second-order superintegrable system in arbitrary dimension is also of this type; irreducibility here means that the endomorphisms K j i = g aj K ia associated with the Killing tensors K ij form an irreducible set, see [KSV19].…”

Section: Introductionmentioning

confidence: 99%

“…For the sake of expositional simplicity, we do not require abundant systems to admit a subset of 2n − 1 functionally independent Killing tensors, asking merely for linearly independent ones. In reference [KSV19] it is proven that systems which do admit 2n−1 functionally independent integrals of the motion lie dense among all systems.…”

Section: Introductionmentioning

confidence: 99%

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Algebraic Conditions for Conformal Superintegrability in Arbitrary Dimension

Kress¹,

Schöbel²,

Vollmer³

2020

Preprint

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Second order conformally superintegrable systems generalise second-order (properly) superintegrable systems. They have been classified, essentially, in dimensions two and (partially) three only. For properly superintegrable systems, a foundation for an algebraic-geometric classification in arbitrary dimension has recently been developed by the authors. The present paper extends this geometric framework to conformally superintegrable systems.Using a rigorously geometric approach, we obtain a set of simple and universal algebraic equations that govern the classification of (the conformal equivalence classes of) second-order conformally superintegrable systems for dimensions three and higher. This sheds new light on the conformal geometry underpinning second-order conformally superintegrable systems. We demonstrate that conformal superintegrability is a conformally invariant concept, on a conformal manifold.For properly superintegrable systems on constant curvature spaces, we show that Stäckel equivalent systems are conformally equivalent with the conformal rescaling given by powers of eigenfunctions of the Laplacian. Invariantly, their equivalence is characterized by a shared density of weight 2. On the n-sphere the conformal scale function satisfies a Laplace eigenvalue equation with quantum number n + 1.

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Section: Introductionmentioning

confidence: 99%

“…We focus our discussion on second-order superintegrable systems whose potential satisfies the Wilczynski equation (introduced in [KSV19])…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Algebraic Conditions for Conformal Superintegrability in Arbitrary Dimension

Kress¹,

Schöbel²,

Vollmer³

2020

Preprint

Self Cite

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“…Second-order superintegrable systems in dimension 2 (2D) are classified [KKPM01,KKM05b,KKM05a,KS18,KPM02]. Their equivalence classes under Stäckel (i.e., conformal) transformations have been characterised [DY06,Kre07].…”

Section: Introductionmentioning

confidence: 99%

Stäckel Equivalence of Non-Degenerate Superintegrable Systems, and Invariant Quadrics

Vollmer

2020

Preprint

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Algebraic Conditions for Conformal Superintegrability in Arbitrary Dimension

Kress,

Schöbel,

Vollmer

2024

Commun. Math. Phys.

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We consider second order (maximally) conformally superintegrable systems and explain how the definition of such a system on a (pseudo-)Riemannian manifold gives rise to a conformally invariant interpretation of superintegrability. Conformal equivalence in this context is a natural extension of the classical (linear) Stäckel transform, originating from the Maupertuis-Jacobi principle. We extend our recently developed algebraic geometric approach for the classification of second order superintegrable systems in arbitrarily high dimension to conformally superintegrable systems, which are presented via conformal scale choices of second order superintegrable systems defined within a conformal geometry. For superintegrable systems on constant curvature spaces, we find that the conformal scales of Stäckel equivalent systems arise from eigenfunctions of the Laplacian and that their equivalence is characterised by a conformal density of weight two. Our approach yields an algebraic equation that governs the classification under conformal equivalence for a prolific class of second order conformally superintegrable systems. This class contains all non-degenerate examples known to date, and is given by a simple algebraic constraint of degree two on a general harmonic cubic form. In this way the yet unsolved classification problem is put into the reach of algebraic geometry and geometric invariant theory. In particular, no obstruction exists in dimension three, and thus the known classification of conformally superintegrable systems is reobtained in the guise of an unrestricted univariate sextic. In higher dimensions, the obstruction is new and has never been revealed by traditional approaches.

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An algebraic geometric classification of superintegrable systems in the Euclidean plane

Cited by 15 publications

References 41 publications

Algebraic Conditions for Conformal Superintegrability in Arbitrary Dimension

Algebraic Conditions for Conformal Superintegrability in Arbitrary Dimension

Stäckel Equivalence of Non-Degenerate Superintegrable Systems, and Invariant Quadrics

Algebraic Conditions for Conformal Superintegrability in Arbitrary Dimension

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