Algebraic Conditions for Conformal Superintegrability in Arbitrary Dimension

Abstract

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.

The sl2(R) coalgebra symmetry and the superintegrable discrete-time systems

“…For the coalgebra approach to these systems, see for example a series of papers by Italian-Spanish school [3][4][5][6][7] culminating in the proof of a Bertrand-like theorem on curved space [12], linked to the so-called Perlick classification [51]. For a different, more geometric perspective on the subject, see the recent classification in [34,35]. Finding a connection between these two approaches and building the discrete-time analogue will be subject of future research.…”