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Relativistic Coulomb Integrals and Zeilberger’s Holonomic Systems Approach II

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Part of the Lecture Notes in Computer Science book series (LNTCS,volume 8372)

Abstract

We derive the recurrence relations for relativistic Coulomb integrals directly from the integral representations with the help of computer algebra methods. In order to manage the computational complexity of this problem, we employ holonomic closure properties in a sophisticated way.

Keywords

  • Coulomb integral
  • holonomic systems approach
  • creative telescoping
  • holonomic closure property
  • operator algebra
  • annihilating ideal

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References

  1. Almkvist, G., Zeilberger, D.: The method of differentiating under the integral sign. Journal of Symbolic Computation 10(6), 571–591 (1990)

    CrossRef  MATH  MathSciNet  Google Scholar 

  2. Buchberger, B.: Ein Algorithmus zum Auffinden der Basiselemente des Rest-klassenrings nach einem nulldimensionalen Polynomideal. PhD thesis, University of Innsbruck, Innsbruck, Austria (1965)

    Google Scholar 

  3. Chyzak, F.: An extension of Zeilberger’s fast algorithm to general holonomic functions. Discrete Mathematics 217(1-3), 115–134 (2000)

    CrossRef  MATH  MathSciNet  Google Scholar 

  4. Coutinho, S.C.: A primer of algebraic D-modules. London Mathematical Society Student Texts, vol. 33. Cambridge University Press (1995)

    Google Scholar 

  5. Gumberidze, A., et al: Quantum electrodynamics in strong electric fields: the ground state Lamb shift in hydrogenlike uranium. Physical Review Letters 94, 223001 (4pp) (2005)

    Google Scholar 

  6. Gumberidze, A., et al: Precision tests of QED in strong fields: experiments on hydrogen- and helium-like uranium. Journal of Physics: Conference Series 58, 87–92 (2007)

    Google Scholar 

  7. Kandri-Rody, A., Weispfenning, V.: Non-commutative Gröbner bases in algebras of solvable type. Journal of Symbolic Computation 9(1), 1–26 (1990)

    CrossRef  MATH  MathSciNet  Google Scholar 

  8. Koutschan, C.: Advanced applications of the holonomic systems approach. PhD thesis, Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Linz, Austria (2009)

    Google Scholar 

  9. Koutschan, C.: A fast approach to creative telescoping. Mathematics in Computer Science 4(2-3), 259–266 (2010)

    CrossRef  MATH  MathSciNet  Google Scholar 

  10. Koutschan, C.: HolonomicFunctions (user’s guide). Technical Report 10-01, RISC Report Series, Johannes Kepler University, Linz, Austria (2010), http://www.risc.jku.at/research/combinat/software/HolonomicFunctions/

  11. Paule, P., Suslov, S.K.: Relativistic Coulomb integrals and Zeilberger’s holonomic systems approach I. In: Schneider, C., Blümlein, J. (eds.) Computer Algebra in Quantum Field Theory. Texts & Monographs in Symbolic Computation, pp. 225–241. Springer, Wien (2013)

    CrossRef  Google Scholar 

  12. Puchkov, A.M.: The method of matrix elements’ calculations for the Dirac equation in the Coulomb field. Journal of Physics B: Atomic, Molecular and Optical 44, 045002 (6pp) (2010)

    Google Scholar 

  13. Puchkov, A.M., Labzovskiĭ, L.N.: Probabilities of forbidden magnetic-dipole transitions in the hydrogen atom and hydrogen-like ions. Optics and Spectroscopy 106(2), 181–186 (2009)

    Google Scholar 

  14. Puchkov, A.M., Labzovskiĭ, L.N.: Parity violation effects in hydrogen atom in forbidden magnetic-dipole transitions. Optics and Spectroscopy 108(5), 713–718 (2010)

    Google Scholar 

  15. Shabaev, V.M.: Generalizations of the virial relations for the Dirac equation in a central field and their applications to the Coulomb field. Journal of Physics B: Atomic, Molecular and Optical 24, 4479–4488 (1991)

    CrossRef  MathSciNet  Google Scholar 

  16. Shabaev, V.M.: Two-time Green’s function method in quantum electrodynamics of high-Z few-electron atoms. Physics Reports 356, 119–228 (2002)

    CrossRef  MATH  Google Scholar 

  17. Shabaev, V.M.: Virial relations for the Dirac equation and their applications to calculations of hydrogen-like atoms. In: Precision Physics of Simple Atomic Systems. Lecture Notes in Physics, vol. 627, pp. 97–113. Springer, Heidelberg (2003)

    CrossRef  Google Scholar 

  18. Shabaev, V.M.: Quantum electrodynamics of heavy ions and atoms: current status and prospects. Physics-Uspekhi 178(11), 1220–1225 (2008) (in Russian)

    Google Scholar 

  19. Suslov, S.K.: Expectation values in relativistic Coulomb problems. Journal of Physics B: Atomic, Molecular and Optical 42, 185003 (8pp) (2009)

    Google Scholar 

  20. Suslov, S.K.: Mathematical structure of relativistic Coulomb integrals. Physical Review A 81, 032110 (2010)

    Google Scholar 

  21. Zeilberger, D.: A fast algorithm for proving terminating hypergeometric identities. Discrete Mathematics 80(2), 207–211 (1990)

    CrossRef  MATH  MathSciNet  Google Scholar 

  22. Zeilberger, D.: A holonomic systems approach to special functions identities. Journal of Computational and Applied Mathematics 32(3), 321–368 (1990)

    CrossRef  MATH  MathSciNet  Google Scholar 

  23. Zeilberger, D.: The method of creative telescoping. Journal of Symbolic Computation 11, 195–204 (1991)

    CrossRef  MATH  MathSciNet  Google Scholar 

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Koutschan, C., Paule, P., Suslov, S.K. (2014). Relativistic Coulomb Integrals and Zeilberger’s Holonomic Systems Approach II. In: Barkatou, M., Cluzeau, T., Regensburger, G., Rosenkranz, M. (eds) Algebraic and Algorithmic Aspects of Differential and Integral Operators. AADIOS 2012. Lecture Notes in Computer Science, vol 8372. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54479-8_6

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  • DOI: https://doi.org/10.1007/978-3-642-54479-8_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-54478-1

  • Online ISBN: 978-3-642-54479-8

  • eBook Packages: Computer ScienceComputer Science (R0)