Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 038, 31 pages      arXiv:1512.09315      https://doi.org/10.3842/SIGMA.2016.038
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications

Bôcher Contractions of Conformally Superintegrable Laplace Equations

Ernest G. Kalnins a, Willard Miller Jr. b and Eyal Subag c
a) Department of Mathematics, University of Waikato, Hamilton, New Zealand
b) School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455, USA
c) Department of Mathematics, Pennsylvania State University, State College, Pennsylvania, 16802 USA

Received January 24, 2016, in final form April 11, 2016; Published online April 19, 2016

Abstract
The explicit solvability of quantum superintegrable systems 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 on constant curvature spaces 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. However, the limits have no satisfactory Lie algebra contraction interpretations for underlying spaces with 1- or 0-dimensional Lie algebras. We show that these systems can be best understood by transforming them to Laplace conformally superintegrable systems, with flat space conformal symmetry group ${\rm SO}(4,{\mathbb C})$, and using ideas introduced in the 1894 thesis of Bôcher to study separable solutions of the wave equation in terms of roots of quadratic forms. We show that Bôcher's prescription for coalescing roots of these forms induces contractions of the conformal algebra $\mathfrak{so}(4,{\mathbb C})$ to itself and yields a mechanism for classifying all Helmholtz superintegrable systems and their limits. In the paper [Acta Polytechnica, to appear, arXiv:1510.09067], we announced our main findings. This paper provides the proofs and more details.

Key words: conformal superintegrability; contractions; Laplace equations.

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