# Separation of variables, superintegrability and BÃ´cher contractions

## Meeting Details

Abstract: Quantum superintegrable systems are exactly solvable quantum 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. The irreducible representations of these algebras yields important information about the eigenvalues and eigenspaces of the quantum systems. Distinct superintegrable systems and their quadratic algebras are related by geometric contractions, induced by generalized InÃ¶nÃ¼-Wigner Lie algebra contractions which have important physical and geometric implications, such as the Askey scheme for obtaining all hypergeometric orthogonal polynomials as limits of Racah/Wilson polynomials. This can all be unified by ideas first introduced in the 1894 thesis of BÃ´cher to study R- separable solutions of the wave equation.