Singular inverse square potential, limit cycles, and self-adjoint extensions

Abstract

We study the radial Schr\"odinger equation for a particle of mass m in the field of a singular attractive $\ensuremath{\alpha}{/r}^{2}$ potential with $2m\ensuremath{\alpha}>1/4.$ 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.