Foundations of Physics

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

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

Article

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).

Keywords

Inverse-square potential Regularization Renormalization  Quantum anomalies Eckart potential 

References

  1. 1.
    Denschlag, J., Umshaus, G., Schmiedmayer, J.: Probing a singular potential with cold atoms: a neutral atom and a charged wire. Phys. Rev. Lett. 81, 737–741 (1998)Google Scholar
  2. 2.
    Bawin, M., Coon, S.A.: Neutral atom and a charged wire: from elastic scattering to absorption. Phys. Rev. A 63, 034701 (2001)CrossRefADSGoogle Scholar
  3. 3.
    Bawin, M.: Electron-bound states in the field of dipolar molecules. Phys. Rev. A 70, 022505 (2004)CrossRefADSGoogle Scholar
  4. 4.
    Denschlag, J., Schmiedmayer, J.: Scattering a neutral atom from a charged wire. Europhys. Lett. 38, 405–410 (1997)CrossRefADSGoogle Scholar
  5. 5.
    Camblong, H.E., Ordonez, C.R.: Anomaly in conformal quantum mechanics: from molecular physics to black holes. Phys. Rev. D 68, 125013 (2003)MathSciNetCrossRefADSGoogle Scholar
  6. 6.
    Efimov, V.: Weakly bound states of three resonantly interacting particles. Sov. J. Nucl. Phys. 12, 589–595 (1971)Google Scholar
  7. 7.
    Bawin, M., Coon, S.A.: Singular inverse square potential, limit cycles, and self-adjoint extensions. Phys. Rev. A 67, 042712 (2003)CrossRefADSGoogle Scholar
  8. 8.
    Beane, S.R., Bedaque, P.F., Childress, L., Kryjevski, A., McGuire, J., van Kolck, U.: Singular potentials and limit cycles. Phys. Rev. A 64, 042103 (2001)CrossRefADSGoogle Scholar
  9. 9.
    Braaten, E., Phillips, D.: Renormalization-group limit cycle for the \(1/r^{2}\) potential. Phys. Rev. A 70, 052111 (2004)CrossRefADSGoogle Scholar
  10. 10.
    Case, K.M.: Singular potentials. Phys. Rev. 80, 797–806 (1950)MathSciNetCrossRefMATHADSGoogle Scholar
  11. 11.
    Landau, L.D., Lifshitz, E.M.: Quantum Mechanics, Course of Theoretical Physics, 3rd edn, pp. 114–117. Pergamon Press, Oxford (1977)Google Scholar
  12. 12.
    Alliluev, S.P.: The problem of collapse to the center in quantum mechanics. Sov. Phys. JETP 34, 8–13 (1972)ADSGoogle Scholar
  13. 13.
    Frank, W.M., Land, D.J., Spector, R.M.: Singular potentials. Rev. Mod. Phys. 43, 36–98 (1971)MathSciNetCrossRefADSGoogle Scholar
  14. 14.
    Parisi, G., Zirilli, F.: Anomalous dimensions in one-dimensional quantum field theory. J. Math. Phys. 14, 243–245 (1973)CrossRefADSGoogle Scholar
  15. 15.
    Radin, C.: Some remarks on the evolution of a Schrödinger particle in an attractive \(1/r^{2}\) potential. J. Math. Phys. 16, 544–547 (1975)MathSciNetCrossRefADSGoogle Scholar
  16. 16.
    Mastalir, R.O.: Theory of Regge poles for \(1/r^{2}\) potentials. Int. J. Math. Phys. 16, 743–748 (1975)MathSciNetCrossRefMATHADSGoogle Scholar
  17. 17.
    Mastalir, R.O.: Theory of Regge poles for \(1/r^{2}\) potentials. II. An exactly solvable example at zero energy. J. Math. Phys. 16, 749–751 (1975).Google Scholar
  18. 18.
    Mastalir, R.O.: Theory of Regge poles for \(1/r^{2}\) potentials. III. An exact solution of Schrödinger’s equation for arbitrary l and E. J. Math. Phys. 16, 752–755 (1975).Google Scholar
  19. 19.
    van Haeringen, H.: Bound states for \(r^{-2}\)-like potentials in one and three dimensions. J. Math. Phys. 19, 2171–2179 (1978)MathSciNetCrossRefADSGoogle Scholar
  20. 20.
    Schwartz, C.: Almost singular potentials. J. Math. Phys. 17, 863–867 (1976)CrossRefADSGoogle Scholar
  21. 21.
    Simon, B.: Essential self-adjointness of Schrödinger operators with singular potentials. Arch. Ration. Mech. Anal. 52, 44–48 (1974)Google Scholar
  22. 22.
    Simander, C.G.: Remarks on Schrödinger operators with strongly singular potentials. Math. Z. 138, 53–70 (1974)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Narnhofer, H.: Quantum theory for \(1/r^{2}\) potentials. Acta Phys. Austriaca 40, 306–332 (1974)MathSciNetGoogle Scholar
  24. 24.
    Coon, S.A., Holstein, B.R.: Anomalies in quantum mechanics: the \(1/r^{2}\) potential. Am. J. Phys. 70, 513–519 (2002)MathSciNetCrossRefMATHADSGoogle Scholar
  25. 25.
    Gupta, K.S., Rajeev, S.G.: Renormalization in quantum mechanics. Phys. Rev. D 48, 5940–5945 (1993)CrossRefADSGoogle Scholar
  26. 26.
    Camblong, H.E., Epele, L.N., Fanchiotti, H., García Canal, C.A.: Renormalization of the inverse square potential. Phys. Rev. Lett. 85, 1590–1593 (2000)CrossRefADSGoogle Scholar
  27. 27.
    Bouaziz, D., Bawin, M.: Regularization of the singular inverse square potential in quantum mechanics with a minimal length. Phys. Rev. A 76, 032112 (2007)CrossRefADSGoogle Scholar
  28. 28.
    Gopalakrishnan, S.: Self-adjointness and the renormalization of singular potentials. Thesis, advised by Loinaz, W., Amherst College, 2006 (unpublished).Google Scholar
  29. 29.
    Essin, A.M., Griffiths, D.J.: Quantum mechanics of the \(1/x^{2}\) potential. Am. J. Phys. 74, 109–117 (2006)CrossRefADSGoogle Scholar
  30. 30.
    Camblong, H.E., Epele, I.N., Fanchiotti, H.: On the inequivalence of renormalization and self-adjoint extensions for quantum singular interactions. Phys. Lett. A 364, 458–464 (2007)MathSciNetCrossRefMATHADSGoogle Scholar
  31. 31.
    Yu Voronin, A.: Singular potentials and annihilation. Phys. Rev. A 67, 062706 (2003)CrossRefADSGoogle Scholar
  32. 32.
    Bouaziz, D., Bawin, M.: Singular inverse-square potential: renormalization and self-adjoint extensions for medium to weak coupling. Phys. Rev. A 89, 022113 (2014)CrossRefADSGoogle Scholar
  33. 33.
    Camblong, H.E., Epele, L.N., Fanchiotti, H., Garcia Canal, C.A.: Quantum Anomaly in Molecular Physics. Phys. Rev. Lett. 87, 220402 (2001)CrossRefADSGoogle Scholar
  34. 34.
    Treiman, S.B., Jackiw, R., Zumino, B., Witten, E.: Current Algebras and Anomalies. World Scientific, Singapore (1985)CrossRefGoogle Scholar
  35. 35.
    Olver, F.W.J.: Asymptotics and Special Functions. Academic Press, New York (1974)Google Scholar
  36. 36.
    Dereziński, J., and Wrochna, M.: Exactly solvable Schrödinger operators. Ann. Henri Poincare 12, 397–418 (2011) pp. 410–411Google Scholar
  37. 37.
    Alhaidari, A. D.: arXiv:1309.1683v3 [quant-ph] (2013), pp. 4–5.
  38. 38.
    Gradshteyn, I. S., and Ryzhik, I. M.: Tables of Integrals, Series, and Products, 7\(^{th}\) ed. (Academic, San Diego, 2007) p. 920.Google Scholar

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

Personalised recommendations