PERTURBED ENERGIES AND EIGENFUNCTIONS OF THE CURVED-SPACE AND FLAT-SPACE ISOTROPIC OSCILLATOR VIA THE RICCATI EQUATION

Perturbation theory via the Riccati equation is applied to the analyti cal solution of the isotropic oscillator in a spherical three-space {1 /R(2)(d(2)/d chi(2)+2 cot chi d/d chi)-l(l+1)/R(2)sin(2) chi-omega(2)R (2)tan(2) chi+ V(chi)/R(2)+2E(n)}Phi(chi)=0, where 1/R is the curvatur e of the space, omega is the vibrational constant, and V(chi) is a per turbation. This perturbation procedure, well adapted to the use of com puter algebra, relies on the solution of the Riccati equation associat ed with the given equation and on the choice of suitable chi-basis fun ctions for expanding the perturbation V(chi) Provided a Sturm-Liouvill e equation can be viewed as a Infeld-Hull factorizable equation with a n additional perturbation, an analytical determination of the perturbe d eigenvalues and eigenfunctions can be carried out, up to a rather hi gh order of the perturbation, by means of simple algebraic manipulatio ns. When expanding V(chi) in a series of (tan chi)(s) terms, the curve d-space isotropic oscillator is relevant to the procedure and closed-f orm expressions of the perturbed energies and functions can be obtaine d. Space-curvature contributions to the isotropic oscillator energies are put in evidence from the comparison of their curved-space expressi on with their flat-space limit. Further applications of the results ar e pointed out.