The Complex-Scaling Coupled-Channel Methods for Atomic and Molecular Resonances in Intense External Fields

  • Shih-I. Chu


Resonance states are characterized by complex energies corresponding to poles of the resolvent operator (E-Ĥ)−1 in the complex-energy plane of a non-physical higher Riemann sheet. Numerous techniques have been developed for computing these poles. One of the most powerful techniques popularized in the last decade is the method known as the complex scaling (coordinate-rotation, complex-coordinate, or dilatation) transformation. 1,2 As a result of the complex scaling transformation, r → re, the eigenvalues corresponding to the bound states of Ĥ stay invariant, while the branch cuts associated with the continuous spectrum of Ĥ are rotated about their respective thresholds by an angle −2α (assuming 0<α<π/2), exposing the complex resonance states in appropriate strips of the complex energy plane. A crucial point from the computational point of view is that the eigen-functions associated with the complex-scaling resonance wave functions are localized, i.e. square integrable. The square integrability led to the extension of well-established bound-state techniques to the determination of resonance energies (ER) and widths (Г) of metastable states.


Vibrational Level Stark Shift Multiphoton Ionization Kinetic Energy Operator Complex Scaling 
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.
    E. Balslev and J.M. Combes, Commun. Math. Phys. 22: 280 (1971)CrossRefGoogle Scholar
  2. A. Aguilar and J.M. Combes, Commun. Math. Phys. 22: 265 (1971)CrossRefGoogle Scholar
  3. B. Simon, Ann. Math. 97: 247 (1973).CrossRefGoogle Scholar
  4. 2.
    Proceedings of the 1978 Sanibel Workshop on Complex Scaling, Intern. J. Quantum Chem. 14: 343–542 (1978).Google Scholar
  5. 3.
    P. Agostini, A. Antonetti, P. Breger, M. Crance, A. Migus, H.G. Muller, and G. Petite, J. Phys. B22: 1971 (1989).Google Scholar
  6. 4.
    R.R. Freeman, P.H. Bucksbaum, H. Milchberg, S. Darrack, D. Schumacher, and M.E. Geusic, Phys. Rev. Lett. 59: 1092 (1987).CrossRefGoogle Scholar
  7. 5.
    S.I. Chu and W.P. Reinhardt, Phys. Rev. Lett. 39: 1195 (1977)CrossRefGoogle Scholar
  8. A. Maquet, S.I. Chu, and W.P. Reinhardt, Phys. Rev. A27: 2946 (1983).CrossRefGoogle Scholar
  9. 6.
    S.I. Chu, Adv. At. Mol. Phys. 21: 197 (1985).Google Scholar
  10. 7.
    S.I. Chu, K. Wang, and E. Layton, J. Opt. Soc. Am. B7: 425 (1990).CrossRefGoogle Scholar
  11. 8.
    J.H. Shirley, Phys. Rev. 138: B979 (1965).CrossRefGoogle Scholar
  12. 9.
    For a recent review on various generalizations of Floquet theories and techniques for the treatment of intense-field multiphoton and nonlinear optical processes, see, S.I. Chu, Adv. Chem. Phys. 73: 739 (1989).Google Scholar
  13. 10.
    S.I. Chu and J. Cooper, Phys. Rev. A32: 2769 (1985).CrossRefGoogle Scholar
  14. 11.
    P. Avan, C. Cohen-Tannoudji, J. Dupont-Roc, and C. Fabre, J. Phys. (Paris) 37: 993 (1976)CrossRefGoogle Scholar
  15. L. Hollberg and J.L. Hall, Phys. Rev. Lett. 53: 230 (1984).CrossRefGoogle Scholar
  16. 12.
    S.I. Chu, Chem. Phys. Lett. 58: 462 (1978).CrossRefGoogle Scholar
  17. 13.
    J. Avron, I. Herbst, and B. Simon, Duke Math. J. 45: 847 (1978).CrossRefGoogle Scholar
  18. 14.
    S.K. Bhattacharya and S.I. Chu, J. Phys. B16: L471 (1983).Google Scholar
  19. 15.
    S.K. Bhattacharya and S.I. Chu, J. Phys. B18: L275 (1985).Google Scholar
  20. 16.
    R.H. Garstang, Rep. Prog. Phys. 40: 105 (1977).Google Scholar
  21. 17.
    G. Wunner, H. Herold, and H. Ruder, J. Phys. B16: 2973 (1983).Google Scholar
  22. 18.
    A. Holle, J. Main, G. Wiebusch, H. Rottke, and K.H. Welge, Phys. Rev. Lett. 61: 161 (1988).CrossRefGoogle Scholar
  23. 19.
    C.H. Iu, G.R. Welch, M.M. Kash, L. Hsu, and D. Kleppner, Phys. Rev. Lett. 63: 1133 (1989).CrossRefGoogle Scholar
  24. 20.
    S.I. Chu, Chem. Phys. Lett. 167: 155 (1990).CrossRefGoogle Scholar
  25. 21.
    C.C. Marston and G.G. Balint-Kurti, J. Chem. Phys. 91: 3571 (1989).CrossRefGoogle Scholar
  26. 22.
    K.K. Datta and S.I. Chu, Chem. Phys. Lett. 87: 357 (1982).CrossRefGoogle Scholar
  27. 23.
    O. Atabek and R. Lefebvre, Chem. Phys. Lett. 84: 233 (1981).CrossRefGoogle Scholar
  28. 24.
    A. Carrington and J. Buttenshaw, Mol. Phys. 44: 267 (1981).CrossRefGoogle Scholar
  29. 25.
    S.I. Chu, C. Laughlin, and K.K. Datta, Chem. Phys. Lett. 98: 476 (1983).CrossRefGoogle Scholar
  30. 26.
    C. Laughlin, K.K. Datta, and S.I. Chu, J. Chem. Phys. 85: 1403 (1986).CrossRefGoogle Scholar
  31. 27.
    C. Cornaggia, D. Normand, J. Morellec, G. Mainfray, and C. Manus, Phys. Rev. A34: 207 (1986).CrossRefGoogle Scholar
  32. 28.
    T.S. Luk and C.K. Rhodes, Phys. Rev. A38: 6180 (1988).CrossRefGoogle Scholar
  33. 29.
    P.H. Bucksbaum, A. Zavriyev, H.G. Muller, and D.W. Schumacher, Phys. Rev. Lett. 64: 1883 (1990).CrossRefGoogle Scholar
  34. 30.
    J.O. Hirschfelder, R.E. Wyatt, and R.D. Coalson, ed., “Lasers, Molecules, and Methods,” Wiley, New York, Adv. Chem. Phys. 73:1–978 (1989).Google Scholar
  35. 31.
    S.I. Chu, J. Chem. Phys. 75: 2215 (1981).CrossRefGoogle Scholar
  36. 32.
    A. Dalgarno and J.T. Lewis, Proc. Roy. soc. A233: 70 (1955).CrossRefGoogle Scholar
  37. 33.
    S.I. Chu, J. Chem. Phys. (in press).Google Scholar

Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • Shih-I. Chu
    • 1
  1. 1.Department of ChemistryUniversity of KansasLawrenceUSA

Personalised recommendations