A renormalized HVT approach for a class of central potential wells

An investigation is carried out to consider a renormalized HVT approach in the context of s-power series expansions for the energy eigenvalues of a particle moving non-relativistically in a central potential well belonging to the class V(r)=−Df(rR), D>0 where f is an appropriate even function of x=r/R and the dimensionless quantity s = (h^2/2μDR)^{1/2} is assumed to be sufficiently small. Previously, the more general class of central potentials of even power series in r is considered and the renormalized recurrence relations from which the expansions of the energy eigenvalues follow, are derived. The s-power series of the renormalized expansion are then given for the initial class of potentials up to third order in s (included) for each energy-level Enl . It is shown that the renormalization parameter Κ enters the coefficients of the renormalized expansion through the state-dependent quantity a_{nl}χ^{1/2} =a_{nl}(1+K ((−d_1D)R^2))^{½}, a_{nl}=(2n+l+32). The question of determining χ is discussed. Our first numerical results are also given and the utility of potentials of the class considered (to which belong the well-known Gaussian and reduced Poschl- Teller potentials) in the study of single–particle states of a Λ in hypernuclei is pointed out.