Quadratic algebras and dynamical symmetry of the Schroedinger equation

As is well known, all exactly solvable problems in quantum mechanics admit a treatment in the language of Lie algebras or groups. It has been shown that the O(2,1) algebra gives rise to those potentials for which the Schroedinger equation reduces to the confluent hypergeometric equation (the harmonic oscillator, Coulomb, and Morse potentials). However such an interpretation is impossible in those cases when the Schroedinger equation reduces to the full hypergeometric equation (potentials of the Poeschl-Teller or Eckart type). Indeed, the spectra of these potentials are quadratic (or reduce to quadratic), while in a Lie algebra the discrete spectrum of any generator can only be linear. In the present work, the authors investigate a quadratic algebra with three generators and simplest choice of nonlinearity, which they call the Jacobi algebra. Within its framework they find a natural interpretation of all exactly solvable problems with a quadratic (discrete or continuous) spectrum in the spirit of the dynamic symmetry idea. The Lie case, corresponding to a potential with a linear spectrum, is obtained by a simple degeneracy (contraction) of the Jacobi algebra