Laplace equations, conformal superintegrability and B\^ocher contractions

TL;DR

This paper explores the classification and contraction of 2D quantum superintegrable systems using conformal Laplace equations and Bôcher's methods, revealing their algebraic structures and geometric implications.

Contribution

It introduces a framework for understanding superintegrable systems through conformal transformations and Bôcher contractions, connecting algebraic and geometric aspects.

Findings

Classification of superintegrable systems via Bôcher contractions

Connection between algebraic contractions and geometric limits

Insights into the structure of quadratic algebras and their symmetries

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 In\"on\"u-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 B\^ocher 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.