Time-Dependent Scattering in Coulombic Few-Body Systems and the Strong Operator Approximation Method

  • Helmut Kröger
Part of the Finite Systems and Multiparticle Dynamics book series (FSMD)


The subject of this chapter is nonrelativistic quantum mechanical scattering in Coulombic few-body systems, using the time-dependent formulation. Our ultimate goal is to find efficient and reliable algorithms for numerical computation of scattering observables, such as phase shifts, cross sections, etc. Historically, the notion of a scattering operator was introduced by Heisenberg(1) and Møller.(2) The so-called Møller wave operator maps asymptotic states onto scattering states. At the time when the Møller wave operator was suggested, there was no mathematical proof of its existence. The first proof was given by Cook,(3) formulated in time-dependent language, for a two-body system interacting via a square integrable potential.


Wave Packet Coulomb Potential Asymptotic State Wave Operator Breakup Reaction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    W. Heisenberg, Z. Phys. 120, 513 (1943).CrossRefGoogle Scholar
  2. 2.
    C. Møller, Danske Vid. Selsk. Mat.-Fys. Medd. 23, 1 (1945).Google Scholar
  3. 3.
    J. M. Cook, J. Math. Phys. 36, 82 (1957).Google Scholar
  4. 4.
    M. Gell-Mann and M. L. Goldberger, Phys. Rev. 91, 398 (1953).CrossRefGoogle Scholar
  5. 5.
    E. O. Alt and W. Sandhas, Chapter 1, this volume.Google Scholar
  6. 6.
    J. Broekhove, L. Lathouwers and P. van Leuven (eds.), Lecture Notes in Physics, Vol. 256 (Springer, Berlin, 1985).Google Scholar
  7. 7.
    V. Mohan and N. Sathyamurthy, Comp. Phys. Rep. 7, 214 (1988).CrossRefGoogle Scholar
  8. 8.
    K. C. Kulander (ed.), Time-Dependent Methods for Quantum Dynamics, Comput. Phys. Comm. 63, 1 (1991).Google Scholar
  9. 9.
    C. Leforestier, R. H. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H. D. Meyer, N. Lipkin, O. Roncero, and R. Kosloff, J. Comput. Phys. 94, 59 (1991).CrossRefGoogle Scholar
  10. 10.
    H. Kröger, Phys. Repts. 210, 45 (1992).CrossRefGoogle Scholar
  11. 11.
    J. Mazur and R. J. Rubin, J. Chem. Phys. 31, 1395 (1959).CrossRefGoogle Scholar
  12. 12.
    E. A. McCullough jr. and R. E. Wyatt, J. Chem. Phys. 51, 1253 (1969).CrossRefGoogle Scholar
  13. 13.
    K. C. Kulander, J. Chem. Phys. 69, 5064 (1978).CrossRefGoogle Scholar
  14. 14.
    K. C. Kulander, Nucl. Phys. A353, 341 (1981).Google Scholar
  15. 15.
    P. M. Agrawal, V. Mohan, and N. Sathyamurthy, Chem. Phys. Lett. 114, 343 (1985).CrossRefGoogle Scholar
  16. 16.
    E. J. Heller, J. Chem. Phys. 68, 2066 (1978).CrossRefGoogle Scholar
  17. 17.
    E. J. Heller, in Potential Energy Surfaces and Dynamics Calculations (D. G. Truhlar, ed.) (Plenum, New York, 1981), p. 103.Google Scholar
  18. 18.
    S. Y. Lee and E. J. Heller, J. Chem. Phys. 71, 4777 (1979).CrossRefGoogle Scholar
  19. 19.
    E. J. Heller and M. J. Davis, J. Chem. Phys. 85, 307 (1981).CrossRefGoogle Scholar
  20. 20.
    G. Drolshagen and E. J. Heller, J. Chem. Phys. 79, 2072 (1983).CrossRefGoogle Scholar
  21. 21.
    M. Batinic, Ž. Bajzer, and H. Kröger, Phys. Rev. C 33, 1187 (1986).CrossRefGoogle Scholar
  22. 22.
    J. Holz and W. Glöckle, Phys. Rev. C 37 1386 (1988).CrossRefGoogle Scholar
  23. 23.
    Z. C. Kuruoglu and F. S. Levin, Phys. Rev. Lett. 64, 1701 (1990).CrossRefGoogle Scholar
  24. 24.
    H. Kröger, Phys. Lett. 135B, 1 (1984).Google Scholar
  25. 25.
    H. Kröger and R. J. Slobodrian, Phys. Rev. C 30, 1390 (1984).CrossRefGoogle Scholar
  26. 26.
    D. Neuhauser, M. Baer, R. Judson, and D. J. Kouri, J. Chem. Phys. 90, 5882 (1989).CrossRefGoogle Scholar
  27. 27.
    D. Neuhauser, M. Baer, R. Judson, and D. J. Kouri, J. Chem. Phys. 93, 312, (1990).CrossRefGoogle Scholar
  28. 28.
    R. Judson, D. J. Kouri, D. Neuhauser, and M. Baer, Phys. Rev. A 42, 351 (1990).CrossRefGoogle Scholar
  29. 29.
    H. Kröger and W. Sandhas, Phys. Rev. Lett. 40, 834 (1978).CrossRefGoogle Scholar
  30. 30.
    M. H. Kalos, Monte Carlo Methods in Quantum Problems, Paris, France, 1982, Proc. Nato Adv. Res. Workshop, Nato ASI Ser. C, Vol. 125 (Reidel, Dordrecht, 1984).Google Scholar
  31. 31.
    K. E. Schmidt and M. H. Kalos, Applications of the Monte Carlo Method in Statistical Physics (K. Binder, ed.) (Springer, Berlin, 1984s).Google Scholar
  32. 32.
    M. H. Kalos and P. A. Whitlock, Monte Carlo Methods (John Wiley, New York, 1986).CrossRefGoogle Scholar
  33. 33.
    H. Kröger, K. J. M. Moriarty, and J. Potvin, Phys. Rev. A 42, 2661 (1990).CrossRefGoogle Scholar
  34. 34.
    W. Glöckle, The Quantum Mechanical Few-Body Problem (Springer, Berlin, 1983).CrossRefGoogle Scholar
  35. 35.
    L. D. Faddeev, Mathematical Aspects of the Three-Body Problem in the Quantum Scattering Theory (Israel Program for Scientific Translation Jerusalem, 1965).Google Scholar
  36. 36.
    W. O. Amrein, J. M. Jauch, and K. B. Sinha, Scattering Theory in Quantum Mechanics, Lecture Notes and Supplements in Physics (Benjamin, Reading, MA., 1977).Google Scholar
  37. 37.
    M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. III: Scattering Theory (Academic Press, New York, 1979).Google Scholar
  38. 38.
    J. Weidmann, Linear Operations in Hilbert Spaces (Springer, New York, 1980).CrossRefGoogle Scholar
  39. 39.
    H. Baumgärtel and M. Wollenberg, Mathematical Scattering Theory (Birkhäuser Verlag, Basel, 1983).Google Scholar
  40. 40.
    H. van Haeringen, Charged-Particle Interactions (Coulomb Press, Leiden, 1985).Google Scholar
  41. 41.
    C. J. Joachain, Quantum Collision Theory (North Holland, Amsterdam, 1975).Google Scholar
  42. 42.
    J. Weidmann, Linear Operations in Hilbert Spaces (Springer, New York, 1980), Theorem 11.13, p. 352.CrossRefGoogle Scholar
  43. 43.
    J. R. Taylor, Scattering Theory (John Wiley, New York, 1972).Google Scholar
  44. 44.
    W. Gordon, Z. Phys. 48, 180 (1928).CrossRefGoogle Scholar
  45. 45.
    J. D. Dollard, Rocky Mountain J. Math. 1, 5 (1971).CrossRefGoogle Scholar
  46. 46.
    J. D. Dollard, Rocky Mountain J. Math. 2, 317 (1972).Google Scholar
  47. 47.
    N. F. Mott and A. S. W. Massey, The Theory of Atomic Collisions (Oxford University Press, 1949).Google Scholar
  48. 48.
    J. D. Dollard, J. Math. Phys. 5, 729 (1964).CrossRefGoogle Scholar
  49. 49.
    G. L. Nutt, J. Math. Phys. 9, 796 (1968).CrossRefGoogle Scholar
  50. 50.
    H. van Haeringen, J. Math. Phys. 17, 995 (1976).CrossRefGoogle Scholar
  51. 51.
    J. Zorbas, Nuov. Cim. Lett. 10, 121 (1974).CrossRefGoogle Scholar
  52. 52.
    J. M. Jauch, Helv. Phys. Acta 31, 127 (1958).Google Scholar
  53. 53.
    J. D. Dollard, J. Math. Phys. 9, 620 (1968).CrossRefGoogle Scholar
  54. 54.
    E. O. ALT, W. Sandhas, and H. Ziegelmann, Nucl. Phys. A445, 429 (1985).Google Scholar
  55. 55.
    I. W. Herbst, Comm. Math. Phys. 35, 181 (1974).CrossRefGoogle Scholar
  56. 56.
    A. M. Veselova, Theor. Math. Phys. 13, 368 (1972).CrossRefGoogle Scholar
  57. 57.
    Z. Bajzer, in Few-Body Nuclear Physics, (G. Pisent, V. Vanzani, and L. Fonda, eds.) (IAEA, Vienna, 1978), p. 365.Google Scholar
  58. 58.
    J. Schwinger, J. Math. Phys. 5, 1606 (1964).CrossRefGoogle Scholar
  59. 59.
    Z. Bajzer, Z. Phys. A278, 97 (1976).Google Scholar
  60. 60.
    A. Messiah, Quantum Mechanics, Vol. I (Elsevier, Amsterdam, 1961).Google Scholar
  61. 61.
    A.A. Samarskii and E. S. Nikolaev, Numerical Methods for Grid Equations (Birkhäuser, Basel, 1989).CrossRefGoogle Scholar
  62. 62.
    C. Moler and C. van Loan, SIAM Rev. 20, 801 (1978).CrossRefGoogle Scholar
  63. 63.
    A. J. Dragt, in Lectures on Nonlinear Orbit Dynamics (Fermilab. 1982), Proceedings of the Conference of High Energy Particle Accelerators, AIP Conf. Proc. No. 87 (R. A. Carrigan and F. R. Huson, eds.) (AIP, New York, 1982), p. 147.Google Scholar
  64. 64.
    E. Forest, Superconducting Super Collider Central Design Group Publication No. SSC-111 (1987).Google Scholar
  65. 65.
    R. S. Varga, Matrix Iterative Analysis (Prentice Hall, Englewood Cliffs, N.J. 1980).Google Scholar
  66. 66.
    H. Kröger, J. Math. Phys. 25, 1875 (1984).CrossRefGoogle Scholar
  67. 67.
    W. Schweiger, W. Plessas, L. P. KOK, and H. van Haeringen, Phys. Rev. C 27, 515 (1983).CrossRefGoogle Scholar
  68. 68.
    R. Girard, H. Kröger, P. Labelle, and Ž. Bajzer, Phys. Rev. A 37, 3195 (1988).CrossRefGoogle Scholar
  69. 69.
    M. Batinic, Ž. Bajzer, and H. Kröger, Phys. Rev. C 38, 2955 (1988).CrossRefGoogle Scholar
  70. 70.
    E. Hadjimichael, Nucl. Phys. A508, 161c (1990).Google Scholar
  71. 71.
    R. Machleidt, K. Holinde, and C. Elster, Phys. Repts. 149, 1 (1987).CrossRefGoogle Scholar
  72. 72.
    R. Machleidt, in Advances in Nuclear Physics, Vol. 19. (J. W. Negele and E. Vogt, eds.) (Plenum, New York, 1989), p. 189.CrossRefGoogle Scholar
  73. 73.
    CH. Hajduk and P. E. Sauer, IX International Conference on Few-Body Problems, (M. J. Moravcsik and F. S. Levin, eds.) University of Oregon, Eugene, 1980), Paper II-16.Google Scholar
  74. 74.
    R. B. Wiringa, Proc. 3rd International Conference on Recent Progress in Many-Body Theory (H. Kümmel and M. Ristic, eds.) (Springer, Berlin, 1983).Google Scholar
  75. 75.
    K. T. KIM, Y. E. KIM, D. J. Klepacki, R. A. Brandenburg, E. P. Harper, R. Machleidt, Phys. Rev. C 38, 2366 (1988).CrossRefGoogle Scholar
  76. 76.
    I. Slaus, Y. Akaishi, and H. Tanaka, Phys. Rev. Lett. 48, 993 (1982).CrossRefGoogle Scholar
  77. 77.
    W. Meier and W. Glöckle, Phys. Lett. B138, 329 (1984).Google Scholar
  78. 78.
    M. Karus, M. Buballa, J. Helten, B. Laumann, R. Melzer, P. Niessen, H. Oswald, G. Rauprich, J. Schulte-Uebbing, and H. Paetz Gen. Schiek, Phys. Rev. C 31, 1112 (1985).CrossRefGoogle Scholar
  79. 79.
    H. Witala, W. Glöckle, and T. Cornelius, Phys. Rev. C 39, 384 (1989).CrossRefGoogle Scholar
  80. 80.
    A. M. Nachabe, R. J. Slobodrian, B. K. Sinha, R. Roy, and H. Kröger, J. Phys. (Paris) 47, 1141 (1986).CrossRefGoogle Scholar
  81. 81.
    H. Kröger and R. J. Slobodrian, Phys. Lett. B144, 19 (1984).Google Scholar
  82. 82.
    V. V. Komarov, A. M. Popova, and V. V. Zatekin, Proc. X. International Conference on Few-Body Problems in Physics, Karlsruhe (B. Zeidnitz, ed.) (North Holland, Amsterdam, 1984), Vol. 2, p. 387.Google Scholar
  83. 83.
    H. Kröger, A. M. Nachabe, and R. J. Slobodrian, Phys. Rev. C 33 1208 (1986).CrossRefGoogle Scholar
  84. 84.
    P. Doleschall, private communication.Google Scholar
  85. 85.
    D. Neuhauser, M. Baer, R. S. Judson, and D. J. Kouri, Comput. Phys. Comm. 63, 460 (1991).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1996

Authors and Affiliations

  • Helmut Kröger
    • 1
  1. 1.Département de PhysiqueUniversité LavalQuébecCanada

Personalised recommendations