Laplace-type equations as conformal superintegrable systems

TL;DR

This paper develops the theory of second-order conformal superintegrable systems, linking Laplace equations with potentials to special functions and symmetry algebras, and classifies 3D Helmholtz systems via conformal group actions.

Contribution

It establishes a foundational framework connecting superintegrable systems, special functions, and conformal symmetry, and reduces classification of 3D Helmholtz systems to orbit classification under conformal group actions.

Findings

Deep connection between superintegrable systems and special functions.

Classification of 3D Helmholtz systems via conformal group orbits.

Unified treatment of Laplace and Helmholtz equations with potentials.

Abstract

We lay out the foundations of the theory of second-order conformal superintegrable systems. Such systems are essentially Laplace equations on a manifold with an added potential: $(\Delta_n+V({\bf x}))\Psi=0$. Distinct families of second-order superintegrable Schr\"odinger (or Helmholtz) systems $(\Delta'_n+V'({\bf x}))\Psi=E\Psi$ can be incorporated into a single Laplace equation. There is a deep connection between most of the special functions of mathematical physics, these Laplace conformally superintegrable systems and their conformal symmetry algebras. Using the theory of the Laplace systems, we show that the problem of classifying all 3D Helmholtz superintegrable systems with nondegenerate potentials, i.e., potentials with a maximal number of independent paprameters, can be reduced to the problem of classifying the orbits of the nonlinear action of the conformal group on a 10-dimensional manifold.