Family of exact solutions for the Coulomb potential perturbed by a polynomial inr

Abstract

A method based on supersymmetric quantum mechanics is given for obtaining exact solutions of the potential V(r)=α/r+${\mathit{p}}_1$r+${\mathit{p}}_2$ ${\mathit{r}}^2$+${\mathit{p}}_3$ ${\mathit{r}}^3$+${\mathit{p}}_4$ ${\mathit{r}}^4$ where α and the p’s are parameters, provided certain relations are satisfied between the parameters. Detailed results are given for three specific cases. The potential in question gives rise to some very interesting shapes (double-well, etc.). The applicability of the shifted 1/N expansion method to such potential shapes is examined by comparing eigenenergies obtained by this method with the exact ones obtained from supersymmetric considerations. It is found that in certain situations, the shifted 1/N expansion method may give poor or erroneous results. Applicability of the proposed method to potentials involving higher powers of r is also discussed.