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Generalized Sturm expansions of the Coulomb Green’s function and two-photon Gordon formulas

  • Atoms, Spectra, Radiation
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Abstract

The radial component of the Coulomb Green’s function (CGF) is written in the form of a double series in Laguerre polynomials (Sturm’s functions in the Coulomb problem), which contains two free parameters α and α′. The obtained result is applicable both in the nonrelativistic case and for the CGF of the squared Dirac equation with a Coulomb potential. The CGF is decomposed into the resonance and potential components (the latter is a smooth function of energy) for α = α′. In the momentum representation, the CGF with the free parameters is written in the form of an expansion in four-dimensional spherical functions. The choice of the parameters α and α ′ in accordance with the specific features of the given problem radically simplifies the calculation of the composite matrix elements for electromagnetic transitions. Closed analytic expressions (in terms of hypergeometric functions) are obtained for the amplitudes of bound-bound and bound-free two-photon transitions in the hydrogen atom from an arbitrary initial state ¦nl〉, which generalize the known (one-photon) Gordon formulas. The dynamic polarizability tensor components αnlm(ω) for an arbitrary n are expressed in terms of the hypergeometric function 2 F 1 depending only on l and \(\tilde \omega \) and through the polynomial functions \(f_{nl} (\tilde \omega )\) of frequency \(\tilde \omega = {{\hbar \omega } \mathord{\left/ {\vphantom {{\hbar \omega } {\left| {E_n } \right|}}} \right. \kern-\nulldelimiterspace} {\left| {E_n } \right|}}\). The Rydberg (n ≫ 1) and threshold (ℏω ∼ ¦ E n¦) asymptotic forms of polarizabilities are investigated.

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References

  1. W. Gordon, Ann. Phys. (Leipzig) 2, 1031 (1929); V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum Electrodynamics (Nauka, Moscow, 1980; Pergamon, Oxford, 1982), Parag. 52; H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One-and Two-Electron Atoms (Academic, New York, 1957; Fizmatgiz, Moscow, 1960), Parag. 63.

    MATH  Google Scholar 

  2. C. Schwartz and T. J. Tiemann, Ann. Phys. (N.Y.) 6, 178 (1959); M. H. Mittleman and F. Wolfe, Phys. Rev. 128, 2686 (1962).

    Google Scholar 

  3. M. Gavrila, Preprint IFA FI-60 (Inst. of Atomic Physics, Bucharest, 1966); Phys. Rev. 163, 147 (1967).

    Google Scholar 

  4. S. I. Vetchinkin and S. V. Khristenko, Chem. Phys. Lett. 1, 437 (1967); S. V. Khristenko and S. I. Vetchinkin, Opt. Spektrosk. 25, 650 (1968).

    Article  ADS  Google Scholar 

  5. Higher Transcendental Functions (Bateman Manuscript Project), Ed. by A. Erdelyi (McGraw-Hill, New York, 1953; Nauka, Moscow, 1973), Vol. 1.

    Google Scholar 

  6. B. A. Zon, N. L. Manakov, and L. P. Rapoport, Zh. Éksp. Teor. Fiz. 55, 924 (1968) [Sov. Phys. JETP 28, 480 (1969)].

    Google Scholar 

  7. M. Gavrila, Z. Phys. A 293, 269 (1979); V. Florescu, Phys. Rev. A 30, 2441 (1984); V. Florescu and A. Cionga, Z. Phys. A 321, 187 (1985).

    Google Scholar 

  8. A. Maquet, Phys. Rev. A 15, 1088 (1977).

    Article  ADS  Google Scholar 

  9. M. A. Preobrazhenskii, Zh. Éksp. Teor. Fiz. 111, 816 (1997) [JETP 84, 448 (1997)].

    Google Scholar 

  10. J. T. Broad, Phys. Rev. A 31, 1494 (1985).

    Article  ADS  Google Scholar 

  11. V. Yakhontov and K. Jungmann, Z. Phys. D 38, 141 (1996).

    Article  Google Scholar 

  12. A. A. Krylovetsky, N. L. Manakov, and S. I. Marmo, Laser Phys. 7, 817 (1997).

    Google Scholar 

  13. A. Costescu, I. Brändus, and N. Mezincescu, J. Phys. B 18, L11 (1985); V. Florescu, S. Pätrascu, and O. Stoican, Phys. Rev. A 36, 2155 (1987).

    Article  ADS  Google Scholar 

  14. E. Karule, J. Phys. B 4, L67 (1971); A. Maquet, Phys. Lett. A 48A, 199 (1974).

    Article  ADS  Google Scholar 

  15. T. A. Marian, Phys. Rev. A 39, 3816 (1989).

    ADS  Google Scholar 

  16. A. I. Ignat’ev, Zh. Éksp. Teor. Fiz. 70, 484 (1976) [Sov. Phys. JETP 43, 250 (1976)].

    Google Scholar 

  17. J. P. Gazeau, J. Math. Phys. 19, 1041 (1978).

    ADS  MathSciNet  Google Scholar 

  18. L. P. Rapoport, B. A. Zon, and N. L. Manakov, Zh. Éksp. Teor. Fiz. 56, 400 (1969) [Sov. Phys. JETP 29, 220 (1969)].

    Google Scholar 

  19. S. Klarsfeld, Lett. Nuovo Cimento 1, 682 (1969); 2, 548 (1969); 3, 395 (1970); V. G. Gorshkov and V. S. Polikanov, Pis’ma Zh. Éksp. Teor. Fiz. 9, 464 (1969) [JETP Lett. 9, 279 (1969)].

    Google Scholar 

  20. M. Gavrila, Lett. Nuovo Cimento 2, 180 (1969); Phys. Rev.A 6, 1360 (1972); Rev. Roum. Phys. 19, 483 (1974).

    Google Scholar 

  21. E. Karule, J. Phys. B 11, 441 (1978); E. Karule and R. H. Pratt, J. Phys. B 24, 1585 (1991); E. Karule, J. Phys. B 18, 220 (1985).

    Article  ADS  Google Scholar 

  22. S. Klarsfeld and A. Maquet, J. Phys. B 12, L553 (1979); Phys. Lett. A 78A, 40 (1980); J. Phys. B 73, 100 (1979).

    Article  ADS  Google Scholar 

  23. A. G. Fainshtein, N. L. Manakov, and S. I. Marmo, Phys. Lett. A 104A, 347 (1984).

    ADS  Google Scholar 

  24. N. L. Manakov, S. I. Marmo, and A. V. Shaposhnikov, in Atoms and Molecules in Strong Field of Laser Radiation, Ed. by F. V. Bunkin and I. I. Tugov (Nauka, Moscow, 1992), p. 87.

    Google Scholar 

  25. E. J. Heller, Phys. Rev. A 12, 1222 (1975).

    Article  ADS  Google Scholar 

  26. N. L. Manakov, A. Maquet, S. I. Marmo, and C. Szymanowski, Phys. Lett. A 237, 234 (1998).

    Article  ADS  Google Scholar 

  27. S. A. Zapryagaev, N. L. Manakov, and V. G. Pal’chikov, Theory of Multiply Charged Ions with One and Two Electrons (Énergoatomizdat, Moscow, 1985), Chap. 2.

    Google Scholar 

  28. N. L. Manakov, V. D. Ovsiannikov, and L. P. Rapoport, Phys. Rep. 141, 319 (1986).

    Article  Google Scholar 

  29. A. Maquet, V. Véniard, and T. A. Marian, J. Phys. B 31, 3743 (1998).

    Article  ADS  Google Scholar 

  30. L. Hostler, J. Math. Phys. 11, 2966 (1970).

    Article  MathSciNet  Google Scholar 

  31. Higher Transcendental Functions (Bateman Manuscript Project), Ed. by A. Erdelyi (McGraw-Hill, New York, 1953; Nauka, Moscow, 1974), Vol. 2.

    Google Scholar 

  32. N. L. Manakov, S. I. Marmo, and A. G. Fainshtein, Teor. Mat. Fiz. 59, 49 (1984).

    MathSciNet  Google Scholar 

  33. E. Arnous, J. Bastian, and A. Maquet, Phys. Rev. A 27, 977 (1983).

    Article  ADS  Google Scholar 

  34. R. Shakeshaft, J. Phys. B 18, L611 (1985); Phys. Rev. A 34, 244 (1986).

    Article  ADS  Google Scholar 

  35. B. A. Zon, N. L. Manakov, and L. P. Rapoport, Yad. Fiz. 15, 508 (1972) [Sov. J. Nucl. Phys. 15, 282 (1972)]; N. L. Manakov, L. P. Rapoport, and S. A. Zapryagaev, Phys. Lett. A 43A, 139 (1973).

    Google Scholar 

  36. J. Schwinger, J. Math. Phys. 5, 1606 (1964).

    Article  MATH  MathSciNet  Google Scholar 

  37. L. C. Hostler, J. Math. Phys. 5, 1235 (1964).

    MathSciNet  Google Scholar 

  38. Higher Transcendental Functions (Bateman Manuscript Project), Ed. by A. Erdelyi (McGraw-Hill, New York, 1955; Nauka, Moscow, 1974), Vol. 3.

    Google Scholar 

  39. R. Papp, J. Phys. A 20, 153 (1987).

    Article  ADS  MathSciNet  Google Scholar 

  40. L. P. Presnyakov and A. M. Urnov, Zh. Éksp. Teor. Fiz. 68, 61 (1975) [Sov. Phys. JETP 41, 31 (1975)].

    Google Scholar 

  41. C. Greene, U. Fano, and G. Strinati, Phys. Rev. A 19, 1485 (1979).

    ADS  Google Scholar 

  42. V. E. Chernov, N. L. Manakov, and A. F. Starace, Eur. Phys. J. D 8, 347 (2000).

    Article  ADS  Google Scholar 

  43. C. Dorman, I. Kucukkara, and J. P. Marangos, Phys. Rev. A 61, 013802 (2000).

    Google Scholar 

  44. G. Z. Zhang, D. W. Tokaryk, and B. P. Stoicheff, Phys. Rev. A 56, 813 (1997).

    ADS  Google Scholar 

  45. N. L. Manakov, A. Maquet, S. I. Marmo, et al., J. Phys. B 32, 3747 (1999).

    Article  ADS  Google Scholar 

  46. B. A. Zon, N. L. Manakov, and L. P. Rapoport, Opt. Spektrosk. 38, 13 (1975) [Opt. Spectrosc. 38, 6 (1975)].

    Google Scholar 

  47. D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum (Nauka, Leningrad, 1975; World Scientific, Singapore, 1988).

    Google Scholar 

  48. N. L. Manakov, V. A. Sviridov, and A. G. Fainshtein, Zh. Éksp. Teor. Fiz. 95, 790 (1989) [Sov. Phys. JETP 68, 451 (1989)].

    Google Scholar 

  49. V. P. Gavrilenko, Zh. Éksp. Teor. Fiz. 90, 857 (1986) [Sov. Phys. JETP 63, 500 (1986)].

    Google Scholar 

  50. I. L. Beigman, Zh. Éksp. Teor. Fiz. 100, 125 (1991) [Sov. Phys. JETP 73, 68 (1991)]; I. L. Beigman, L. A. Bureeva, and R. H. Pratt, J. Phys. B 27, 5833 (1994).

    Google Scholar 

  51. S. V. Khristenko and S. I. Vetchinkin, Opt. Spektrosk. 26, 310 (1969).

    Google Scholar 

  52. A. M. Perelomov and V. S. Popov, Zh. Éksp. Teor. Fiz. 50, 179 (1966) [Sov. Phys. JETP 23, 118 (1966)].

    Google Scholar 

  53. N. E. Firsova and A. I. Sherstyuk, Teor. Mat. Fiz. 81, 59 (1989).

    MathSciNet  Google Scholar 

  54. R. Szmytkowski, J. Phys. A 31, 4963 (1998); 31, 7415 (1998).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  55. A. I. Sherstyuk, Opt. Spektrosk. 87, 765 (1999) [Opt. Spectrosc. 87, 696 (1999)].

    Google Scholar 

  56. B. R. Junker, Adv. At. Mol. Phys. 18, 208 (1982); N. Moiseyev, Phys. Rep. 302, 211 (1998).

    Google Scholar 

  57. H. A. Yamani and W. P. Reinhardt, Phys. Rev. A 11, 1144 (1975).

    Article  ADS  Google Scholar 

  58. A. Maquet, S. I. Chu, and W. P. Reinhardt, Phys. Rev. A 27, 2946 (1983).

    Article  ADS  Google Scholar 

  59. R. M. Potvliege and R. Shakeshaft, in Atoms in Intense Laser Fields, Ed. by M. Gavrila (Academic, New York, 1992), p. 373.

    Google Scholar 

  60. R. Gersbacher and J. T. Broad, J. Phys. B 23, 365 (1990).

    Article  ADS  Google Scholar 

  61. N. L. Manakov, S. I. Marmo, E. A. Pronin, and A. F. Starace, in Abstracts of Annual Conference of American Physical Society, DAMOP’2000, Storrs: Connecticut, USA, 14–17 June, 2000 [Bull. Amer. Phys. Soc. 45, 118 (2000)].

  62. L. J. Dubé and J. T. Broad, J. Phys. B 23, 1711 (1990).

    ADS  Google Scholar 

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  1. Voronezh State University, Universitetskaya pl. 1, Voronezh, 394693, Russia

    A. A. Krylovetsky, N. L. Manakov & S. I. Marmo

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__________

Translated from Zhurnal Éksperimental’no\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{l} \) i Teoretichesko\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{l} \) Fiziki, Vol. 119, No. 1, 2001, pp. 45–70.

Original Russian Text Copyright © 2001 by Krylovetsky, Manakov, Marmo.

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Krylovetsky, A.A., Manakov, N.L. & Marmo, S.I. Generalized Sturm expansions of the Coulomb Green’s function and two-photon Gordon formulas. J. Exp. Theor. Phys. 92, 37–60 (2001). https://doi.org/10.1134/1.1348460

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