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

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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 0–1/4. 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).

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Notes

  1. See, for example, Eq. (44) in Ref. [24], Eq. (3.19) in Ref. [25], Eq. (9) in Ref. [26], and Eq. (38) in Ref. [29].

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Acknowledgments

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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  1. A. D. Alhaidari

    Present address: , Bahçelievler Mahallesi, Fatih Caddesi, Rauf Orbay Sokak, Çinar Apartmani 1, No. 17, Yalova, Turkey

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  1. Saudi Center for Theoretical Physics, P. O. Box 32741, Jeddah, 21438, Saudi Arabia

    A. D. Alhaidari

  2. Physics Department, King Fahd University of Petroleum & Minerals, Dhahran, 31261, Saudi Arabia

    A. D. Alhaidari

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Alhaidari, A.D. Renormalization of the Strongly Attractive Inverse Square Potential: Taming the Singularity. Found Phys 44, 1049–1058 (2014). https://doi.org/10.1007/s10701-014-9828-7

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