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

Abstract: We study the radial Schr\"{o}dinger equation for a particle of mass $m$ in the field of the inverse-square potential $\alpha/r^{2}$ in the medium-weak-coupling region, i.e., with $-1/4\leq2m\alpha/\hbar^{2}\leq3/4$. By using the renormalization method of Beane \textit{et} \textit{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 $2m\alpha/\hbar^{2}=-1/4$ is in contradiction with some results in the literature.