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The singular inverse square potential, limit cycles and self-adjoint extensions

M. Bawin·S. A. Coon
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Abstract

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.

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M. Bawin, S. A. Coon. 2003-04-23. The singular inverse square potential, limit cycles and self-adjoint extensions. https://doi.org/10.1103/physreva.67.042712

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