Singular inverse-square potential: Renormalization and self-adjoint extensions for medium to weak coupling

Abstract

We study the radial Schr\"odinger equation for a particle of mass $m$ in the field of the inverse-square potential $\ensuremath{\alpha}/{r}^{2}$ in the medium--weak-coupling region, i.e., with $\ensuremath{-}1/4\ensuremath{\le}2m\ensuremath{\alpha}/{\ensuremath{\hbar}}^{2}\ensuremath{\le}3/4$. By using the renormalization method of Beane et al., with two regularization potentials, a spherical square well and a spherical $\ensuremath{\delta}$ shell, we illustrate that the procedure of renormalization is independent of the choice of the regularization counterterm. We show that, in the aforementioned range of the coupling constant $\ensuremath{\alpha}$, there exists at most one bound state, in complete agreement with the method of self-adjoint extensions. We explicitly show that this bound state is due to the attractive square-well and delta-function counterterms present in the renormalization scheme. Our result for $2m\ensuremath{\alpha}/{\ensuremath{\hbar}}^{2}=\ensuremath{-}1/4$ is in contradiction with some results in the literature.