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

Journal Article · Tue Apr 01 04:00:00 EST 2003 · Physical Review. A

DOI:https://doi.org/10.1103/PhysRevA.67.042712· OSTI ID:20633985

Bawin, M; Coon, S A ^[1]

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1. Universite de Liege, Institut de Physique B5, Sart Tilman, 4000 Liege 1 (Belgium)

We study the radial Schroedinger equation for a particle of mass m in the field of a singular attractive {alpha}/r{sup 2} potential with 2m{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.

• OSTI ID: 20633985
• Journal Information: Physical Review. A, Journal Name: Physical Review. A Journal Issue: 4 Vol. 67; ISSN 1050-2947; ISSN PLRAAN
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
74 ATOMIC AND MOLECULAR PHYSICS
ANIONS
ATOMS
BOSE-EINSTEIN CONDENSATION
DIPOLES
FABRICATION
HAMILTONIANS
LIMIT CYCLE
MOLECULES
NANOSTRUCTURES
OPTICS
POTENTIALS
QUANTUM MECHANICS
RENORMALIZATION
SCHROEDINGER EQUATION
THREE-BODY PROBLEM