Foundations of Physics

, Volume 44, Issue 10, pp 1049–1058

Renormalization of the Strongly Attractive Inverse Square Potential: Taming the Singularity


DOI: 10.1007/s10701-014-9828-7

Cite this article as:
Alhaidari, A.D. Found Phys (2014) 44: 1049. doi:10.1007/s10701-014-9828-7


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 0–1/4. 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).


Inverse-square potential Regularization Renormalization  Quantum anomalies Eckart potential 

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Saudi Center for Theoretical PhysicsJeddahSaudi Arabia
  2. 2.Physics DepartmentKing Fahd University of Petroleum & MineralsDhahranSaudi Arabia
  3. 3.YalovaTurkey

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