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

Understand

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.