  We study the radial Schroedinger equation for a particle in the field of a
singular inverse square attractive potential. This potential is relevant to the
fabrication of nanoscale atom optical devices, is said to be the potential
describing the dipole-bound anions of polar molecules, and is the effective
potential underlying the universal behavior of three-body systems in nuclear
physics and atomic physics, including aspects of Bose-Einstein condensates,
first described by Efimov. New results in three-body physical systems motivate
the present investigation. Using the regularization method of Beane et al., we
show that the corresponding ``renormalization group flow'' equation can be
solved analytically. We find that it exhibits a limit cycle behavior and has
infinitely many branches. We show that a physical meaning for self-adjoint
extensions of the Hamiltonian arises naturally in this framework.
