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We study the radial Schr\"{o}dinger equation for a particle of mass $m$ inthe field of the inverse-square potential $\alpha/r^{2}$ in themedium-weak-coupling region, i.e., with $-1/4\leq2m\alpha/\hbar^{2}\leq3/4$. Byusing the renormalization method of Beane \textit{et} \textit{al.,}with tworegularization potentials, a spherical square well and a spherical $\delta$shell, we illustrate that the procedure of renormalization is independent ofthe choice of the regularization counterterm. We show that, in theaforementioned range of the coupling constant $\alpha$, there exists at mostone bound state, in complete agreement with the method of self-adjointextensions. We explicitly show that this bound state is due to the attractivesquare-well and delta-function counterterms present in the renormalizationscheme. Our result for $2m\alpha/\hbar^{2}=-1/4$ is in contradiction with someresults in the literature.

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