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

TL;DR

This paper addresses the quantum anomaly of the inverse square potential by introducing a renormalization scheme that regularizes the singularity, ensuring a well-defined, self-adjoint Hamiltonian with bounded oscillations and a stable energy spectrum.

Contribution

It proposes a renormalization approach using the Eckart potential to tame the singularity in the inverse square potential, ensuring physical solutions and a bounded spectrum.

Findings

Regularization with Eckart potential yields discrete bound states.

Ensures Hamiltonian is self-adjoint and physically meaningful.

Prevents unbounded oscillations near the origin.

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).