Mutual integrability, quadratic algebras, and dynamical symmetry

Abstract

The concept of mutually integrable dynamical variables is proposed. This concept leads to the quadratic Askey-Wilson algebra QAW(3) which is the dynamical symmetry algebra for all problems where the most general “classical” polynoials arise. In classical mechanics the algebra of the same structure describes the time evolution of dynamical variables in terms of elementary functions. We apply the special case of QAW(3)—Jacobi algebra—to describe the dynamical symmetry of exactly solvable potentials and to resolve the “Manning mystery”—the intimate relation between classical and quantum exactly solvable potentials.