Quadratic algebras and dynamics in curved spaces. II. The Kepler problem

The symmetry aspects of the Kepler problem in a space of constant negative curvature are considered. It is shown that the algebra of the hidden symmetry reduces to the quadratic Racah algebra $QR(3)$, and this makes it possible to express the coefficients of the overlapping of the wave functions in the spherical and parabolic coordinates in terms of Wilson–Racah polynomials. It is shown that the dynamical symmetry algebra that generates the spectrum is the quadratic Jacobi algebra $QJ(3)$. Its ladder operators permit explicit construction of wave functions in the coordinate representation with the ground state as the starting point.