| Popis: |
Perturbation theory via the Riccati equation is applied to the analytical solution of the isotropic oscillator in a spherical three-space {1/${\mathit{R}}^{2}$(${\mathit{d}}^{2}$/d${\mathrm{\ensuremath{\chi}}}^{2}$+2 cot\ensuremath{\chi}d/d\ensuremath{\chi})-l(l+1) /${\mathit{R}}^{2}$${\mathrm{sin}}^{2}$\ensuremath{\chi}-${\mathrm{\ensuremath{\omega}}}^{2}$${\mathit{R}}^{2}$${\mathrm{tan}}^{2}$\ensuremath{\chi}+V(\ensuremath{\chi}) /${\mathit{R}}^{2}$+2${\mathit{E}}_{\mathit{n}}$}\ensuremath{\Phi}(\ensuremath{\chi})=0,where 1/R is the curvature of the space, \ensuremath{\omega} is the vibrational constant, and V(\ensuremath{\chi}) is a perturbation. This perturbation procedure, well adapted to the use of computer algebra, relies on the solution of the Riccati equation associated with the given equation and on the choice of suitable \ensuremath{\chi}-basis functions for expanding the perturbation V(\ensuremath{\chi}). Provided a Sturm-Liouville equation can be viewed as a Infeld-Hull factorizable equation with an additional perturbation, an analytical determination of the perturbed eigenvalues and eigenfunctions can be carried out, up to a rather high order of the perturbation, by means of simple algebraic manipulations. When expanding V(\ensuremath{\chi}) in a series of (tan\ensuremath{\chi}${)}^{\mathit{s}}$ terms, the curved-space isotropic oscillator is relevant to the procedure and closed-form expressions of the perturbed energies and functions can be obtained. Space-curvature contributions to the isotropic oscillator energies are put in evidence from the comparison of their curved-space expression with their flat-space limit. Further applications of the results are pointed out. \textcopyright{} 1996 The American Physical Society. |
