Laplace equations, conformal superintegrability and Bôcher contractions

Abstract

Quantum superintegrable systems are solvable eigenvalue problems. Their solvability is due to symmetry, but the symmetry is often "hidden". The symmetry generators of 2nd order superintegrable systems in 2 dimensions close under commutation to define quadratic algebras, a generalization of Lie algebras. Distinct systems and their algebras are related by geometric limits, induced by generalized Inönü-Wigner Lie algebra contractions of the symmetry algebras of the underlying spaces. These have physical/geometric implications, such as the Askey scheme for hypergeometric orthogonal polynomials. The systems can be best understood by transforming them to Laplace conformally superintegrable systems and using ideas introduced in the 1894 thesis of Bôcher to study separable solutions of the wave equation. The contractions can be subsumed into contractions of the conformal algebra $so(4,C)$ to itself. Here we announce main findings, with detailed classifications in papers under preparation.

10 pages, 2 figures

References in corpus (6)

• Classification of quantum superintegrable systems with quadratic integrals on two dimensional manifolds
• Invariant classification of second-order conformally flat superintegrable systems
• Quantum Super-Integrable Systems as Exactly Solvable Models
• Models for Quadratic Algebras Associated with Second Order Superintegrable Systems in 2D
• Invariant Classification and Limits of Maximally Superintegrable Systems in 3D
• Wilson polynomials/functions and intertwining operators for the generic quantum superintegrable system on the 2-sphere

Cited by in corpus (1)

• Bôcher Contractions of Conformally Superintegrable Laplace Equations