Foundations of Physics

, Volume 44, Issue 10, pp 1049–1058 | Cite as

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

  • A. D. Alhaidari


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 



The generous support provided by the Saudi Center for Theoretical Physics (SCTP) is highly appreciated. We also acknowledge partial support by King Fahd University of Petroleum and Minerals under group projects number RG1109-1 & RG1109-2. We are grateful to the anonymous Referee for pointing out some errors in the original version of the paper and for suggesting changes that resulted in improving the presentation.


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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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