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
Almkvist, G., Zeilberger, D.: The method of differentiating under the integral sign. Journal of Symbolic Computation 10(6), 571–591 (1990)
Buchberger, B.: Ein Algorithmus zum Auffinden der Basiselemente des Rest-klassenrings nach einem nulldimensionalen Polynomideal. PhD thesis, University of Innsbruck, Innsbruck, Austria (1965)
Chyzak, F.: An extension of Zeilberger’s fast algorithm to general holonomic functions. Discrete Mathematics 217(1-3), 115–134 (2000)
Coutinho, S.C.: A primer of algebraic D-modules. London Mathematical Society Student Texts, vol. 33. Cambridge University Press (1995)
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)
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)
Kandri-Rody, A., Weispfenning, V.: Non-commutative Gröbner bases in algebras of solvable type. Journal of Symbolic Computation 9(1), 1–26 (1990)
Koutschan, C.: Advanced applications of the holonomic systems approach. PhD thesis, Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Linz, Austria (2009)
Koutschan, C.: A fast approach to creative telescoping. Mathematics in Computer Science 4(2-3), 259–266 (2010)
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/
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)
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)
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)
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)
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)
Shabaev, V.M.: Two-time Green’s function method in quantum electrodynamics of high-Z few-electron atoms. Physics Reports 356, 119–228 (2002)
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)
Shabaev, V.M.: Quantum electrodynamics of heavy ions and atoms: current status and prospects. Physics-Uspekhi 178(11), 1220–1225 (2008) (in Russian)
Suslov, S.K.: Expectation values in relativistic Coulomb problems. Journal of Physics B: Atomic, Molecular and Optical 42, 185003 (8pp) (2009)
Suslov, S.K.: Mathematical structure of relativistic Coulomb integrals. Physical Review A 81, 032110 (2010)
Zeilberger, D.: A fast algorithm for proving terminating hypergeometric identities. Discrete Mathematics 80(2), 207–211 (1990)
Zeilberger, D.: A holonomic systems approach to special functions identities. Journal of Computational and Applied Mathematics 32(3), 321–368 (1990)
Zeilberger, D.: The method of creative telescoping. Journal of Symbolic Computation 11, 195–204 (1991)
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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
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