Renormalization of the strongly attractive inverse square potential: Taming the singularity

Abstract

Quantum anomalies in the inverse square potential are well known and widely investigated. Most prominent is the unbounded increase in oscillations of the particle's state as it approaches the origin when the attractive coupling parameter is greater than the critical value of 1/4. Due to this unphysical divergence in oscillations, we are proposing that the interaction gets screened at short distances making the coupling parameter acquire an effective (renormalized) value that falls within the weak range 0 to 1/4. This prevents the oscillations form growing without limit giving a lower bound to the energy spectrum and forcing the Hamiltonian of the system to be self-adjoint. Technically, this translates into a regularization scheme whereby the inverse square potential is replaced near the origin by another that has the same singularity but with a weak coupling strength. Here, we take the Eckart as the regularizing potential and obtain the corresponding solutions (discrete bound states and continuum scattering states).

10 pages, 1 figure, some typos and errors are corrected in this revised version

References in corpus (3)

• Anomaly in conformal quantum mechanics: From molecular physics to black holes
• Singular inverse-square potential: renormalization and self-adjoint extensions for medium to weak coupling
• On the Inequivalence of Renormalization and Self-Adjoint Extensions for Quantum Singular Interactions

Cited by in corpus (8)

• Singular inverse square potential in coordinate space with a minimal length
• Point-Particle Effective Field Theory I: Classical Renormalization and the Inverse-Square Potential
• Quantum propagation across cosmological singularities
• Time of falling of a quantum particle into an inverse square potential
• Bound states and the potential parameter spectrum
• J-matrix method of scattering for inverse-square singular potential with supercritical coupling I. Theory
• Five-parameter potential box with inverse square singular boundaries
• J-matrix method of scattering for inverse-square singular potentials with supercritical coupling II. Regularization