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

Abstract

We study the radial Schroedinger equation for a particle of mass m in the field of a singular attractive α/r 2 potential with 2mα > 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.

Renormalization-group limit cycle for the1∕r2potential

“…In two previous studies of the 1/r 2 potential using renormalization theory, the regularization potential was chosen to be a spherical square well [11,12]:…”

“…The simplest example is the quantum mechanics of a particle in a potential whose long-range behavior is 1/r 2 . This problem has been studied previously within the renormalization group framework by two different groups using a spherical square-well regularization potential [11,12]. Beane et al [11] showed that there are infinitely many choices for the coupling constant of the square-well potential, including a continuous function of the shortdistance cutoff R and a log-periodic function of R with discontinuities which corresponds to an RG limit cycle.…”

“…Beane et al [11] showed that there are infinitely many choices for the coupling constant of the square-well potential, including a continuous function of the shortdistance cutoff R and a log-periodic function of R with discontinuities which corresponds to an RG limit cycle. Bawin and Coon [12] presented a closed-form solution for the coupling constant that is log-periodic, which suggests that the choice with the RG limit cycle is in some sense natural.…”

Renormalization in conformal quantum mechanics

“…Finally, in the core renormalization framework , the strength ℵ of the regularizing core is promoted to a running coupling ℵ = ℵ(a), while the conformal coupling λ remains fixed [22,23]. In this framework, our unified description leads to…”

“…(17) provides the characteristic logperiodic running coupling ℵ(a) of the core-the celebrated limit cycle for the renormalization group of the three-body problem [22,23,24]. For d = 3, l = 0, and zero energy,…”