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Few-Body Systems

, 61:3 | Cite as

The Hankel Transform of the Hulthén Green’s Function

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Abstract

It is shown that the Hankel transform of the s-wave Hulthén physical Green’s function satisfies a second-order differential equation. This equation is solved by applying the proper boundary conditions in association with the properties of the special functions of mathematics to get a closed form expression for the same. The Hankel transform of the physical Green’s function is exploited to extract off-shell solutions and Half- and off-shell T-matrices in the maximal reduced form. The check on our expressions with particular emphasis on their limiting behaviour is made and is found in order. The bound state spectrums of n–p and \(\hbox {n-C}^{12}\) systems are computed by exploiting the associated Jost function and found excellent agreement with experimental results.

Notes

References

  1. 1.
    U. Laha, Off-shell Jost solution for the Hulthén potential. Few-Body Syst. 59, 68 (2018)ADSCrossRefGoogle Scholar
  2. 2.
    U. Laha, An integral transform of Green’s function, off-shell Jost solution and T-matrix for Coulomb–Yamaguchi potential in coordinate representation. Pramana J. Phys. 72, 457–472 (2009)ADSCrossRefGoogle Scholar
  3. 3.
    U. Laha, J. Bhoi, Off-shell Jost solutions for Coulomb and Coulomb-like interactions in all partial waves. J. Math. Phys. 54, 013514 (2013)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    M.G. Fuda, J.S. Whiting, Generalization of the Jost function and its application to off-shell scattering. Phys. Rev. C 8, 1255–1261 (1973)ADSCrossRefGoogle Scholar
  5. 5.
    U. Laha, J. Bhoi, Integral transform of the Coulomb Green’s function by the Hankel function and off-shell scattering. Phys. Rev. C 88, 064001 (2013)ADSCrossRefGoogle Scholar
  6. 6.
    R.G. Newton, Scattering Theory of Waves and Particles (Mc-Graw-Hill, New York, 1982)CrossRefGoogle Scholar
  7. 7.
    L.J. Slater, Generalized Hypergeometric Functions (Cambridge University Press, London, 1966)zbMATHGoogle Scholar
  8. 8.
    A. Erdeyli, Higher Transcendental Functions, vol. 1 (Mc-Graw-Hill, New York, 1953)Google Scholar
  9. 9.
    A.W. Babister, Transcendental Functions Satisfying Non-homogeneous Linear Differential Equations (MacMillan, New York, 1967)zbMATHGoogle Scholar
  10. 10.
    H. van Haeringen, Charged Particle Interactions- Theory and Formulas (The Coulomb Press, Leyden, 1985)Google Scholar
  11. 11.
    U. Laha, C. Bhattacharyya, K. Roy, B. Talukdar, Hamiltonian hierarchy and the Hulthén potential. Phys. Rev. C 38, 558–560 (1988)ADSCrossRefGoogle Scholar
  12. 12.
    U. Laha, J. Bhoi, Hadron-Hadron Scattering Within the Separable Model of Interactions (Scholars’ Press, Beau Bassin, 2018)Google Scholar
  13. 13.
    J. Bhoi, U. Laha, Hamiltonian hierarchy and n-p scattering. J. Phys. G Nucl. Part. Phys. 40, 045107 (2013)ADSCrossRefGoogle Scholar
  14. 14.
    U. Laha, J. Bhoi, Comparative study of the energy dependent and independent two-nucleon interactions: a supersymmetric approach. Int. J. Mod. Phys. E 23, 1450039 (2014)ADSCrossRefGoogle Scholar
  15. 15.
    J. Bhoi, U. Laha, Supersymmetry-generated Jost functions and nucleon-nucleon scattering phase shifts. Phys. Atomic Nucl. 78, 831–834 (2015)ADSCrossRefGoogle Scholar
  16. 16.
    U. Laha, J. Bhoi, Higher partial-wave potentials from supersymmetry-inspired factorization and nucleon-nucleus elastic scattering. Phys. Rev. C 91, 034614 (2015)ADSCrossRefGoogle Scholar
  17. 17.
    J. Bhoi, U. Laha, Nucleon-nucleon scattering phase shifts via supersymmetry and the phase function method. Braz. J. Phys. 46, 129–132 (2016)ADSCrossRefGoogle Scholar
  18. 18.
    U. Laha, J. Bhoi, Parameterization of the Nuclear Hulthe’n Potentials. Phys. Atomic Nucl. 79, 62–66 (2016)ADSCrossRefGoogle Scholar
  19. 19.
    J. Bhoi, U. Laha, Elastic scattering of light nuclei through a simple potential model. Phys. Atomic Nucl. 79, 370–374 (2016)ADSCrossRefGoogle Scholar
  20. 20.
    J. Bhoi, U. Laha, Hulthen potential model for \(\upalpha -\upalpha \) and \(\upalpha \)-\(\text{ He }^{3}\) elastic scattering. Pramana J. Phys. 88, 42 (2017)ADSCrossRefGoogle Scholar
  21. 21.
    J. Bhoi, U. Laha, Supersymmetry inspired low energy \(\upalpha \)-p elastic scattering phases. Theor. Math. Phys. 190, 69–76 (2017)CrossRefGoogle Scholar
  22. 22.
    J. Bhoi, R. Upadhyay, U. Laha, Parameterization of the nuclear Hulthén potential for the nucleus-nucleus elastic scattering. Commun. Theor. Phys. 69, 203–210 (2018)ADSCrossRefGoogle Scholar
  23. 23.
    U. Laha, M. Majumder, J. Bhoi, Volterra integral equation-factorization method and nucleus-nucleus elastic scattering. Pramana J. Phys. 90, 48 (2018)ADSCrossRefGoogle Scholar
  24. 24.
    L.G. Arnold, A.D. MacKellar, Study of equivalent local potentials obtained from separable two-nucleon interactions. Phys. Rev. C 3, 1095–1103 (1971)ADSCrossRefGoogle Scholar
  25. 25.
    W. Tornow, R.L. Walter, R.C. Byrd, Phase-shift-analysis approach to elastic neutron scattering from \(^{12}\)C between 9 and 12 MeV. J. Phys. G Nucl. Phys 11, 379–391 (1985)ADSCrossRefGoogle Scholar
  26. 26.
    W.N. Balley, Generalised Hypergeometric Series (Cambridge University Press, London, 1935)Google Scholar
  27. 27.
    H. van Haeringen, R. van Wageningen, Analytic T matrices for Coulomb plus rational separable potentials. J. Math. Phys. 7, 1441–1452 (1975)MathSciNetCrossRefGoogle Scholar
  28. 28.
    U. Laha, An integral transform of the Coulomb Green’s function and off-shell Scattering. J. Phys. A Math. Gen. 38, 6141–6146 (2005)ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    O.P. Bahethi, M.G. Fuda, The T Matrix for the Hulthén Potential. J. Math. Phys. 12, 2076–2080 (1971)ADSCrossRefGoogle Scholar
  30. 30.
    A.N. Mitra, in Advances in Nuclear Physics, Vol. 3, ed. by M. Baranger, E. Vogt (Plenum Press, 1969) , p. 1–70Google Scholar
  31. 31.
    W. van Dijk, M. Razavy, The dependence of the photodisintegration cross section on the off-shell T-matrix. Nucl. Phys. A 204, 412–426 (1973)ADSCrossRefGoogle Scholar

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© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of PhysicsNational Institute of Technology, JamshedpurJamshedpurIndia

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