Bôcher Contractions of Conformally Superintegrable Laplace Equations

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 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. 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 B^ocher to study separable solutions of the wave equation in terms of roots of quadratic forms. We show that B^ocher'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.