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

We study the radial Schr"{o}dinger equation for a particle of mass $m$ in the field of the inverse-square potential $\alpha \mathrm{/}{r}^2$ in the medium-weak-coupling region, i.e., with $-1\mathrm{/}4\le 2m\alpha \mathrm{/}{\mathrm{\hslash }}^2\le 3\mathrm{/}4$. By using the renormalization method of Beane $\mathit{\text{et}}$ $\mathit{\text{al.,}}$with two regularization potentials, a spherical square well and a spherical $\delta$ 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 $\alpha$, 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 $2m\alpha \mathrm{/}{\mathrm{\hslash }}^2=-1\mathrm{/}4$ is in contradiction with some results in the literature.