Continuation of the wave function for higher angular momentum states to D dimensions I. The generalized Schwartz expansion

Annals of Physics

Volumn 251, Issue 2, 1996, Pages 266-318

  Dunn, M ^a   Watson, D K ^a  

^a UNIVERSITY OF OKLAHOMA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030295421     PISSN: 00034916     EISSN: None     Source Type: Journal    
DOI: 10.1006/aphy.1996.0114     Document Type: Article

References (415)

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24. 2 are held fixed. As in [10], angular momentum and dimensionality enter into the one-electron Schrödinger equation in equivalent and interchangeable ways [44], and so the 1/n expansion is also equivalent to dimensional perturbation theory about the D → ∞ limit.
25. 2 are held fixed. As in [10], angular momentum and dimensionality enter into the one-electron Schrödinger equation in equivalent and interchangeable ways [44], and so the 1/n expansion is also equivalent to dimensional perturbation theory about the D → ∞ limit.
26. 2 are held fixed. As in [10], angular momentum and dimensionality enter into the one-electron Schrödinger equation in equivalent and interchangeable ways [44], and so the 1/n expansion is also equivalent to dimensional perturbation theory about the D → ∞ limit.
27. 2 are held fixed. As in [10], angular momentum and dimensionality enter into the one-electron Schrödinger equation in equivalent and interchangeable ways [44], and so the 1/n expansion is also equivalent to dimensional perturbation theory about the D → ∞ limit.
28. 2 are held fixed. As in [10], angular momentum and dimensionality enter into the one-electron Schrödinger equation in equivalent and interchangeable ways [44], and so the 1/n expansion is also equivalent to dimensional perturbation theory about the D → ∞ limit.
29. 2 are held fixed. As in [10], angular momentum and dimensionality enter into the one-electron Schrödinger equation in equivalent and interchangeable ways [44], and so the 1/n expansion is also equivalent to dimensional perturbation theory about the D → ∞ limit.
30. 2 are held fixed. As in [10], angular momentum and dimensionality enter into the one-electron Schrödinger equation in equivalent and interchangeable ways [44], and so the 1/n expansion is also equivalent to dimensional perturbation theory about the D → ∞ limit.
31. 2 are held fixed. As in [10], angular momentum and dimensionality enter into the one-electron Schrödinger equation in equivalent and interchangeable ways [44], and so the 1/n expansion is also equivalent to dimensional perturbation theory about the D → ∞ limit.
32. 2 are held fixed. As in [10], angular momentum and dimensionality enter into the one-electron Schrödinger equation in equivalent and interchangeable ways [44], and so the 1/n expansion is also equivalent to dimensional perturbation theory about the D → ∞ limit.
33. 2 are held fixed. As in [10], angular momentum and dimensionality enter into the one-electron Schrödinger equation in equivalent and interchangeable ways [44], and so the 1/n expansion is also equivalent to dimensional perturbation theory about the D → ∞ limit.
34. 2 are held fixed. As in [10], angular momentum and dimensionality enter into the one-electron Schrödinger equation in equivalent and interchangeable ways [44], and so the 1/n expansion is also equivalent to dimensional perturbation theory about the D → ∞ limit.
35. 2 are held fixed. As in [10], angular momentum and dimensionality enter into the one-electron Schrödinger equation in equivalent and interchangeable ways [44], and so the 1/n expansion is also equivalent to dimensional perturbation theory about the D → ∞ limit.
36. 2 are held fixed. As in [10], angular momentum and dimensionality enter into the one-electron Schrödinger equation in equivalent and interchangeable ways [44], and so the 1/n expansion is also equivalent to dimensional perturbation theory about the D → ∞ limit.
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47. 23. A. D. Dolgov and V. S. Popov, Phys. Lett. B 86 (1979), 185; H. J. de Vega, Commun. Math. Phys. 70 (1979), 29; A. V. Kudinov and M. A. Smondyrev, Czech. J. Phys. B 32 (1982), 556; Teor. Mat. Fiz. 56 (1983), 357 (Theor. Math. Phys. 56 (1983), 871); J. Avan and H. J. de Vega, Nucl. Phys. B 224 (1983), 61; Phys. Rev. D 29 (1984), 2891; 29 (1984), 2904; J. Avan, Nucl. Phys. B 237 (1984), 159; U. Sukhatme and T. Imbo, Phys. Rev. D 28 (1983), 418; R. S. Gangopadhyay, G. Ghosh, and B. Dutta-Roy, Phys. Rev. A 30 (1984), 594; A. Chatterjee, J. Phys. A 18 (1985), 2403; Phys. Rev. A 34 (1986), 2470; M. Jameel, J. Phys. A 19 (1986), 1967; B. Roy, Phys. Rev. A 34 (1986), 5108; A. Z. Tang and F. T. Chan, Phys. Rev. A 35 (1987), 911; R. Dutt and Y. P. Varshni, J. Phys. B 20 (1987), 2437; Y. P. Varshni, Phys. Rev. A 36 (1987), 3009; 38 (1988), 1595; 40 (1989), 2180; B. Roy and R. Roychoudhury, J. Phys. A 20 (1987), 3051; R. Roychoudhury and Y. P. Varshni, Phys. Rev. A 37 (1988), 2309; R. Sever and C. Tezcan, Phys. Rev. A 36 (1987), 1045; E. Papp, Phys. Rev. A 36 (1987), 3550; 38 (1988), 2158; H. Christiansen, L. N. Epele, H. Franchiotti, and C. A. Garcia Canal, Phys. Rev. A 40 (1989), 1760; S. M. Ikhdair and R. Sever, Z. Phys. D 28 (1993), 1.
48. 23. A. D. Dolgov and V. S. Popov, Phys. Lett. B 86 (1979), 185; H. J. de Vega, Commun. Math. Phys. 70 (1979), 29; A. V. Kudinov and M. A. Smondyrev, Czech. J. Phys. B 32 (1982), 556; Teor. Mat. Fiz. 56 (1983), 357 (Theor. Math. Phys. 56 (1983), 871); J. Avan and H. J. de Vega, Nucl. Phys. B 224 (1983), 61; Phys. Rev. D 29 (1984), 2891; 29 (1984), 2904; J. Avan, Nucl. Phys. B 237 (1984), 159; U. Sukhatme and T. Imbo, Phys. Rev. D 28 (1983), 418; R. S. Gangopadhyay, G. Ghosh, and B. Dutta-Roy, Phys. Rev. A 30 (1984), 594; A. Chatterjee, J. Phys. A 18 (1985), 2403; Phys. Rev. A 34 (1986), 2470; M. Jameel, J. Phys. A 19 (1986), 1967; B. Roy, Phys. Rev. A 34 (1986), 5108; A. Z. Tang and F. T. Chan, Phys. Rev. A 35 (1987), 911; R. Dutt and Y. P. Varshni, J. Phys. B 20 (1987), 2437; Y. P. Varshni, Phys. Rev. A 36 (1987), 3009; 38 (1988), 1595; 40 (1989), 2180; B. Roy and R. Roychoudhury, J. Phys. A 20 (1987), 3051; R. Roychoudhury and Y. P. Varshni, Phys. Rev. A 37 (1988), 2309; R. Sever and C. Tezcan, Phys. Rev. A 36 (1987), 1045; E. Papp, Phys. Rev. A 36 (1987), 3550; 38 (1988), 2158; H. Christiansen, L. N. Epele, H. Franchiotti, and C. A. Garcia Canal, Phys. Rev. A 40 (1989), 1760; S. M. Ikhdair and R. Sever, Z. Phys. D 28 (1993), 1.
49. 23. A. D. Dolgov and V. S. Popov, Phys. Lett. B 86 (1979), 185; H. J. de Vega, Commun. Math. Phys. 70 (1979), 29; A. V. Kudinov and M. A. Smondyrev, Czech. J. Phys. B 32 (1982), 556; Teor. Mat. Fiz. 56 (1983), 357 (Theor. Math. Phys. 56 (1983), 871); J. Avan and H. J. de Vega, Nucl. Phys. B 224 (1983), 61; Phys. Rev. D 29 (1984), 2891; 29 (1984), 2904; J. Avan, Nucl. Phys. B 237 (1984), 159; U. Sukhatme and T. Imbo, Phys. Rev. D 28 (1983), 418; R. S. Gangopadhyay, G. Ghosh, and B. Dutta-Roy, Phys. Rev. A 30 (1984), 594; A. Chatterjee, J. Phys. A 18 (1985), 2403; Phys. Rev. A 34 (1986), 2470; M. Jameel, J. Phys. A 19 (1986), 1967; B. Roy, Phys. Rev. A 34 (1986), 5108; A. Z. Tang and F. T. Chan, Phys. Rev. A 35 (1987), 911; R. Dutt and Y. P. Varshni, J. Phys. B 20 (1987), 2437; Y. P. Varshni, Phys. Rev. A 36 (1987), 3009; 38 (1988), 1595; 40 (1989), 2180; B. Roy and R. Roychoudhury, J. Phys. A 20 (1987), 3051; R. Roychoudhury and Y. P. Varshni, Phys. Rev. A 37 (1988), 2309; R. Sever and C. Tezcan, Phys. Rev. A 36 (1987), 1045; E. Papp, Phys. Rev. A 36 (1987), 3550; 38 (1988), 2158; H. Christiansen, L. N. Epele, H. Franchiotti, and C. A. Garcia Canal, Phys. Rev. A 40 (1989), 1760; S. M. Ikhdair and R. Sever, Z. Phys. D 28 (1993), 1.
50. 23. A. D. Dolgov and V. S. Popov, Phys. Lett. B 86 (1979), 185; H. J. de Vega, Commun. Math. Phys. 70 (1979), 29; A. V. Kudinov and M. A. Smondyrev, Czech. J. Phys. B 32 (1982), 556; Teor. Mat. Fiz. 56 (1983), 357 (Theor. Math. Phys. 56 (1983), 871); J. Avan and H. J. de Vega, Nucl. Phys. B 224 (1983), 61; Phys. Rev. D 29 (1984), 2891; 29 (1984), 2904; J. Avan, Nucl. Phys. B 237 (1984), 159; U. Sukhatme and T. Imbo, Phys. Rev. D 28 (1983), 418; R. S. Gangopadhyay, G. Ghosh, and B. Dutta-Roy, Phys. Rev. A 30 (1984), 594; A. Chatterjee, J. Phys. A 18 (1985), 2403; Phys. Rev. A 34 (1986), 2470; M. Jameel, J. Phys. A 19 (1986), 1967; B. Roy, Phys. Rev. A 34 (1986), 5108; A. Z. Tang and F. T. Chan, Phys. Rev. A 35 (1987), 911; R. Dutt and Y. P. Varshni, J. Phys. B 20 (1987), 2437; Y. P. Varshni, Phys. Rev. A 36 (1987), 3009; 38 (1988), 1595; 40 (1989), 2180; B. Roy and R. Roychoudhury, J. Phys. A 20 (1987), 3051; R. Roychoudhury and Y. P. Varshni, Phys. Rev. A 37 (1988), 2309; R. Sever and C. Tezcan, Phys. Rev. A 36 (1987), 1045; E. Papp, Phys. Rev. A 36 (1987), 3550; 38 (1988), 2158; H. Christiansen, L. N. Epele, H. Franchiotti, and C. A. Garcia Canal, Phys. Rev. A 40 (1989), 1760; S. M. Ikhdair and R. Sever, Z. Phys. D 28 (1993), 1.
51. 23. A. D. Dolgov and V. S. Popov, Phys. Lett. B 86 (1979), 185; H. J. de Vega, Commun. Math. Phys. 70 (1979), 29; A. V. Kudinov and M. A. Smondyrev, Czech. J. Phys. B 32 (1982), 556; Teor. Mat. Fiz. 56 (1983), 357 (Theor. Math. Phys. 56 (1983), 871); J. Avan and H. J. de Vega, Nucl. Phys. B 224 (1983), 61; Phys. Rev. D 29 (1984), 2891; 29 (1984), 2904; J. Avan, Nucl. Phys. B 237 (1984), 159; U. Sukhatme and T. Imbo, Phys. Rev. D 28 (1983), 418; R. S. Gangopadhyay, G. Ghosh, and B. Dutta-Roy, Phys. Rev. A 30 (1984), 594; A. Chatterjee, J. Phys. A 18 (1985), 2403; Phys. Rev. A 34 (1986), 2470; M. Jameel, J. Phys. A 19 (1986), 1967; B. Roy, Phys. Rev. A 34 (1986), 5108; A. Z. Tang and F. T. Chan, Phys. Rev. A 35 (1987), 911; R. Dutt and Y. P. Varshni, J. Phys. B 20 (1987), 2437; Y. P. Varshni, Phys. Rev. A 36 (1987), 3009; 38 (1988), 1595; 40 (1989), 2180; B. Roy and R. Roychoudhury, J. Phys. A 20 (1987), 3051; R. Roychoudhury and Y. P. Varshni, Phys. Rev. A 37 (1988), 2309; R. Sever and C. Tezcan, Phys. Rev. A 36 (1987), 1045; E. Papp, Phys. Rev. A 36 (1987), 3550; 38 (1988), 2158; H. Christiansen, L. N. Epele, H. Franchiotti, and C. A. Garcia Canal, Phys. Rev. A 40 (1989), 1760; S. M. Ikhdair and R. Sever, Z. Phys. D 28 (1993), 1.
52. 23. A. D. Dolgov and V. S. Popov, Phys. Lett. B 86 (1979), 185; H. J. de Vega, Commun. Math. Phys. 70 (1979), 29; A. V. Kudinov and M. A. Smondyrev, Czech. J. Phys. B 32 (1982), 556; Teor. Mat. Fiz. 56 (1983), 357 (Theor. Math. Phys. 56 (1983), 871); J. Avan and H. J. de Vega, Nucl. Phys. B 224 (1983), 61; Phys. Rev. D 29 (1984), 2891; 29 (1984), 2904; J. Avan, Nucl. Phys. B 237 (1984), 159; U. Sukhatme and T. Imbo, Phys. Rev. D 28 (1983), 418; R. S. Gangopadhyay, G. Ghosh, and B. Dutta-Roy, Phys. Rev. A 30 (1984), 594; A. Chatterjee, J. Phys. A 18 (1985), 2403; Phys. Rev. A 34 (1986), 2470; M. Jameel, J. Phys. A 19 (1986), 1967; B. Roy, Phys. Rev. A 34 (1986), 5108; A. Z. Tang and F. T. Chan, Phys. Rev. A 35 (1987), 911; R. Dutt and Y. P. Varshni, J. Phys. B 20 (1987), 2437; Y. P. Varshni, Phys. Rev. A 36 (1987), 3009; 38 (1988), 1595; 40 (1989), 2180; B. Roy and R. Roychoudhury, J. Phys. A 20 (1987), 3051; R. Roychoudhury and Y. P. Varshni, Phys. Rev. A 37 (1988), 2309; R. Sever and C. Tezcan, Phys. Rev. A 36 (1987), 1045; E. Papp, Phys. Rev. A 36 (1987), 3550; 38 (1988), 2158; H. Christiansen, L. N. Epele, H. Franchiotti, and C. A. Garcia Canal, Phys. Rev. A 40 (1989), 1760; S. M. Ikhdair and R. Sever, Z. Phys. D 28 (1993), 1.
53. 23. A. D. Dolgov and V. S. Popov, Phys. Lett. B 86 (1979), 185; H. J. de Vega, Commun. Math. Phys. 70 (1979), 29; A. V. Kudinov and M. A. Smondyrev, Czech. J. Phys. B 32 (1982), 556; Teor. Mat. Fiz. 56 (1983), 357 (Theor. Math. Phys. 56 (1983), 871); J. Avan and H. J. de Vega, Nucl. Phys. B 224 (1983), 61; Phys. Rev. D 29 (1984), 2891; 29 (1984), 2904; J. Avan, Nucl. Phys. B 237 (1984), 159; U. Sukhatme and T. Imbo, Phys. Rev. D 28 (1983), 418; R. S. Gangopadhyay, G. Ghosh, and B. Dutta-Roy, Phys. Rev. A 30 (1984), 594; A. Chatterjee, J. Phys. A 18 (1985), 2403; Phys. Rev. A 34 (1986), 2470; M. Jameel, J. Phys. A 19 (1986), 1967; B. Roy, Phys. Rev. A 34 (1986), 5108; A. Z. Tang and F. T. Chan, Phys. Rev. A 35 (1987), 911; R. Dutt and Y. P. Varshni, J. Phys. B 20 (1987), 2437; Y. P. Varshni, Phys. Rev. A 36 (1987), 3009; 38 (1988), 1595; 40 (1989), 2180; B. Roy and R. Roychoudhury, J. Phys. A 20 (1987), 3051; R. Roychoudhury and Y. P. Varshni, Phys. Rev. A 37 (1988), 2309; R. Sever and C. Tezcan, Phys. Rev. A 36 (1987), 1045; E. Papp, Phys. Rev. A 36 (1987), 3550; 38 (1988), 2158; H. Christiansen, L. N. Epele, H. Franchiotti, and C. A. Garcia Canal, Phys. Rev. A 40 (1989), 1760; S. M. Ikhdair and R. Sever, Z. Phys. D 28 (1993), 1.
54. 23. A. D. Dolgov and V. S. Popov, Phys. Lett. B 86 (1979), 185; H. J. de Vega, Commun. Math. Phys. 70 (1979), 29; A. V. Kudinov and M. A. Smondyrev, Czech. J. Phys. B 32 (1982), 556; Teor. Mat. Fiz. 56 (1983), 357 (Theor. Math. Phys. 56 (1983), 871); J. Avan and H. J. de Vega, Nucl. Phys. B 224 (1983), 61; Phys. Rev. D 29 (1984), 2891; 29 (1984), 2904; J. Avan, Nucl. Phys. B 237 (1984), 159; U. Sukhatme and T. Imbo, Phys. Rev. D 28 (1983), 418; R. S. Gangopadhyay, G. Ghosh, and B. Dutta-Roy, Phys. Rev. A 30 (1984), 594; A. Chatterjee, J. Phys. A 18 (1985), 2403; Phys. Rev. A 34 (1986), 2470; M. Jameel, J. Phys. A 19 (1986), 1967; B. Roy, Phys. Rev. A 34 (1986), 5108; A. Z. Tang and F. T. Chan, Phys. Rev. A 35 (1987), 911; R. Dutt and Y. P. Varshni, J. Phys. B 20 (1987), 2437; Y. P. Varshni, Phys. Rev. A 36 (1987), 3009; 38 (1988), 1595; 40 (1989), 2180; B. Roy and R. Roychoudhury, J. Phys. A 20 (1987), 3051; R. Roychoudhury and Y. P. Varshni, Phys. Rev. A 37 (1988), 2309; R. Sever and C. Tezcan, Phys. Rev. A 36 (1987), 1045; E. Papp, Phys. Rev. A 36 (1987), 3550; 38 (1988), 2158; H. Christiansen, L. N. Epele, H. Franchiotti, and C. A. Garcia Canal, Phys. Rev. A 40 (1989), 1760; S. M. Ikhdair and R. Sever, Z. Phys. D 28 (1993), 1.
55. 23. A. D. Dolgov and V. S. Popov, Phys. Lett. B 86 (1979), 185; H. J. de Vega, Commun. Math. Phys. 70 (1979), 29; A. V. Kudinov and M. A. Smondyrev, Czech. J. Phys. B 32 (1982), 556; Teor. Mat. Fiz. 56 (1983), 357 (Theor. Math. Phys. 56 (1983), 871); J. Avan and H. J. de Vega, Nucl. Phys. B 224 (1983), 61; Phys. Rev. D 29 (1984), 2891; 29 (1984), 2904; J. Avan, Nucl. Phys. B 237 (1984), 159; U. Sukhatme and T. Imbo, Phys. Rev. D 28 (1983), 418; R. S. Gangopadhyay, G. Ghosh, and B. Dutta-Roy, Phys. Rev. A 30 (1984), 594; A. Chatterjee, J. Phys. A 18 (1985), 2403; Phys. Rev. A 34 (1986), 2470; M. Jameel, J. Phys. A 19 (1986), 1967; B. Roy, Phys. Rev. A 34 (1986), 5108; A. Z. Tang and F. T. Chan, Phys. Rev. A 35 (1987), 911; R. Dutt and Y. P. Varshni, J. Phys. B 20 (1987), 2437; Y. P. Varshni, Phys. Rev. A 36 (1987), 3009; 38 (1988), 1595; 40 (1989), 2180; B. Roy and R. Roychoudhury, J. Phys. A 20 (1987), 3051; R. Roychoudhury and Y. P. Varshni, Phys. Rev. A 37 (1988), 2309; R. Sever and C. Tezcan, Phys. Rev. A 36 (1987), 1045; E. Papp, Phys. Rev. A 36 (1987), 3550; 38 (1988), 2158; H. Christiansen, L. N. Epele, H. Franchiotti, and C. A. Garcia Canal, Phys. Rev. A 40 (1989), 1760; S. M. Ikhdair and R. Sever, Z. Phys. D 28 (1993), 1.
56. 23. A. D. Dolgov and V. S. Popov, Phys. Lett. B 86 (1979), 185; H. J. de Vega, Commun. Math. Phys. 70 (1979), 29; A. V. Kudinov and M. A. Smondyrev, Czech. J. Phys. B 32 (1982), 556; Teor. Mat. Fiz. 56 (1983), 357 (Theor. Math. Phys. 56 (1983), 871); J. Avan and H. J. de Vega, Nucl. Phys. B 224 (1983), 61; Phys. Rev. D 29 (1984), 2891; 29 (1984), 2904; J. Avan, Nucl. Phys. B 237 (1984), 159; U. Sukhatme and T. Imbo, Phys. Rev. D 28 (1983), 418; R. S. Gangopadhyay, G. Ghosh, and B. Dutta-Roy, Phys. Rev. A 30 (1984), 594; A. Chatterjee, J. Phys. A 18 (1985), 2403; Phys. Rev. A 34 (1986), 2470; M. Jameel, J. Phys. A 19 (1986), 1967; B. Roy, Phys. Rev. A 34 (1986), 5108; A. Z. Tang and F. T. Chan, Phys. Rev. A 35 (1987), 911; R. Dutt and Y. P. Varshni, J. Phys. B 20 (1987), 2437; Y. P. Varshni, Phys. Rev. A 36 (1987), 3009; 38 (1988), 1595; 40 (1989), 2180; B. Roy and R. Roychoudhury, J. Phys. A 20 (1987), 3051; R. Roychoudhury and Y. P. Varshni, Phys. Rev. A 37 (1988), 2309; R. Sever and C. Tezcan, Phys. Rev. A 36 (1987), 1045; E. Papp, Phys. Rev. A 36 (1987), 3550; 38 (1988), 2158; H. Christiansen, L. N. Epele, H. Franchiotti, and C. A. Garcia Canal, Phys. Rev. A 40 (1989), 1760; S. M. Ikhdair and R. Sever, Z. Phys. D 28 (1993), 1.
57. 23. A. D. Dolgov and V. S. Popov, Phys. Lett. B 86 (1979), 185; H. J. de Vega, Commun. Math. Phys. 70 (1979), 29; A. V. Kudinov and M. A. Smondyrev, Czech. J. Phys. B 32 (1982), 556; Teor. Mat. Fiz. 56 (1983), 357 (Theor. Math. Phys. 56 (1983), 871); J. Avan and H. J. de Vega, Nucl. Phys. B 224 (1983), 61; Phys. Rev. D 29 (1984), 2891; 29 (1984), 2904; J. Avan, Nucl. Phys. B 237 (1984), 159; U. Sukhatme and T. Imbo, Phys. Rev. D 28 (1983), 418; R. S. Gangopadhyay, G. Ghosh, and B. Dutta-Roy, Phys. Rev. A 30 (1984), 594; A. Chatterjee, J. Phys. A 18 (1985), 2403; Phys. Rev. A 34 (1986), 2470; M. Jameel, J. Phys. A 19 (1986), 1967; B. Roy, Phys. Rev. A 34 (1986), 5108; A. Z. Tang and F. T. Chan, Phys. Rev. A 35 (1987), 911; R. Dutt and Y. P. Varshni, J. Phys. B 20 (1987), 2437; Y. P. Varshni, Phys. Rev. A 36 (1987), 3009; 38 (1988), 1595; 40 (1989), 2180; B. Roy and R. Roychoudhury, J. Phys. A 20 (1987), 3051; R. Roychoudhury and Y. P. Varshni, Phys. Rev. A 37 (1988), 2309; R. Sever and C. Tezcan, Phys. Rev. A 36 (1987), 1045; E. Papp, Phys. Rev. A 36 (1987), 3550; 38 (1988), 2158; H. Christiansen, L. N. Epele, H. Franchiotti, and C. A. Garcia Canal, Phys. Rev. A 40 (1989), 1760; S. M. Ikhdair and R. Sever, Z. Phys. D 28 (1993), 1.
58. 23. A. D. Dolgov and V. S. Popov, Phys. Lett. B 86 (1979), 185; H. J. de Vega, Commun. Math. Phys. 70 (1979), 29; A. V. Kudinov and M. A. Smondyrev, Czech. J. Phys. B 32 (1982), 556; Teor. Mat. Fiz. 56 (1983), 357 (Theor. Math. Phys. 56 (1983), 871); J. Avan and H. J. de Vega, Nucl. Phys. B 224 (1983), 61; Phys. Rev. D 29 (1984), 2891; 29 (1984), 2904; J. Avan, Nucl. Phys. B 237 (1984), 159; U. Sukhatme and T. Imbo, Phys. Rev. D 28 (1983), 418; R. S. Gangopadhyay, G. Ghosh, and B. Dutta-Roy, Phys. Rev. A 30 (1984), 594; A. Chatterjee, J. Phys. A 18 (1985), 2403; Phys. Rev. A 34 (1986), 2470; M. Jameel, J. Phys. A 19 (1986), 1967; B. Roy, Phys. Rev. A 34 (1986), 5108; A. Z. Tang and F. T. Chan, Phys. Rev. A 35 (1987), 911; R. Dutt and Y. P. Varshni, J. Phys. B 20 (1987), 2437; Y. P. Varshni, Phys. Rev. A 36 (1987), 3009; 38 (1988), 1595; 40 (1989), 2180; B. Roy and R. Roychoudhury, J. Phys. A 20 (1987), 3051; R. Roychoudhury and Y. P. Varshni, Phys. Rev. A 37 (1988), 2309; R. Sever and C. Tezcan, Phys. Rev. A 36 (1987), 1045; E. Papp, Phys. Rev. A 36 (1987), 3550; 38 (1988), 2158; H. Christiansen, L. N. Epele, H. Franchiotti, and C. A. Garcia Canal, Phys. Rev. A 40 (1989), 1760; S. M. Ikhdair and R. Sever, Z. Phys. D 28 (1993), 1.
59. 23. A. D. Dolgov and V. S. Popov, Phys. Lett. B 86 (1979), 185; H. J. de Vega, Commun. Math. Phys. 70 (1979), 29; A. V. Kudinov and M. A. Smondyrev, Czech. J. Phys. B 32 (1982), 556; Teor. Mat. Fiz. 56 (1983), 357 (Theor. Math. Phys. 56 (1983), 871); J. Avan and H. J. de Vega, Nucl. Phys. B 224 (1983), 61; Phys. Rev. D 29 (1984), 2891; 29 (1984), 2904; J. Avan, Nucl. Phys. B 237 (1984), 159; U. Sukhatme and T. Imbo, Phys. Rev. D 28 (1983), 418; R. S. Gangopadhyay, G. Ghosh, and B. Dutta-Roy, Phys. Rev. A 30 (1984), 594; A. Chatterjee, J. Phys. A 18 (1985), 2403; Phys. Rev. A 34 (1986), 2470; M. Jameel, J. Phys. A 19 (1986), 1967; B. Roy, Phys. Rev. A 34 (1986), 5108; A. Z. Tang and F. T. Chan, Phys. Rev. A 35 (1987), 911; R. Dutt and Y. P. Varshni, J. Phys. B 20 (1987), 2437; Y. P. Varshni, Phys. Rev. A 36 (1987), 3009; 38 (1988), 1595; 40 (1989), 2180; B. Roy and R. Roychoudhury, J. Phys. A 20 (1987), 3051; R. Roychoudhury and Y. P. Varshni, Phys. Rev. A 37 (1988), 2309; R. Sever and C. Tezcan, Phys. Rev. A 36 (1987), 1045; E. Papp, Phys. Rev. A 36 (1987), 3550; 38 (1988), 2158; H. Christiansen, L. N. Epele, H. Franchiotti, and C. A. Garcia Canal, Phys. Rev. A 40 (1989), 1760; S. M. Ikhdair and R. Sever, Z. Phys. D 28 (1993), 1.
60. 23. A. D. Dolgov and V. S. Popov, Phys. Lett. B 86 (1979), 185; H. J. de Vega, Commun. Math. Phys. 70 (1979), 29; A. V. Kudinov and M. A. Smondyrev, Czech. J. Phys. B 32 (1982), 556; Teor. Mat. Fiz. 56 (1983), 357 (Theor. Math. Phys. 56 (1983), 871); J. Avan and H. J. de Vega, Nucl. Phys. B 224 (1983), 61; Phys. Rev. D 29 (1984), 2891; 29 (1984), 2904; J. Avan, Nucl. Phys. B 237 (1984), 159; U. Sukhatme and T. Imbo, Phys. Rev. D 28 (1983), 418; R. S. Gangopadhyay, G. Ghosh, and B. Dutta-Roy, Phys. Rev. A 30 (1984), 594; A. Chatterjee, J. Phys. A 18 (1985), 2403; Phys. Rev. A 34 (1986), 2470; M. Jameel, J. Phys. A 19 (1986), 1967; B. Roy, Phys. Rev. A 34 (1986), 5108; A. Z. Tang and F. T. Chan, Phys. Rev. A 35 (1987), 911; R. Dutt and Y. P. Varshni, J. Phys. B 20 (1987), 2437; Y. P. Varshni, Phys. Rev. A 36 (1987), 3009; 38 (1988), 1595; 40 (1989), 2180; B. Roy and R. Roychoudhury, J. Phys. A 20 (1987), 3051; R. Roychoudhury and Y. P. Varshni, Phys. Rev. A 37 (1988), 2309; R. Sever and C. Tezcan, Phys. Rev. A 36 (1987), 1045; E. Papp, Phys. Rev. A 36 (1987), 3550; 38 (1988), 2158; H. Christiansen, L. N. Epele, H. Franchiotti, and C. A. Garcia Canal, Phys. Rev. A 40 (1989), 1760; S. M. Ikhdair and R. Sever, Z. Phys. D 28 (1993), 1.
61. 23. A. D. Dolgov and V. S. Popov, Phys. Lett. B 86 (1979), 185; H. J. de Vega, Commun. Math. Phys. 70 (1979), 29; A. V. Kudinov and M. A. Smondyrev, Czech. J. Phys. B 32 (1982), 556; Teor. Mat. Fiz. 56 (1983), 357 (Theor. Math. Phys. 56 (1983), 871); J. Avan and H. J. de Vega, Nucl. Phys. B 224 (1983), 61; Phys. Rev. D 29 (1984), 2891; 29 (1984), 2904; J. Avan, Nucl. Phys. B 237 (1984), 159; U. Sukhatme and T. Imbo, Phys. Rev. D 28 (1983), 418; R. S. Gangopadhyay, G. Ghosh, and B. Dutta-Roy, Phys. Rev. A 30 (1984), 594; A. Chatterjee, J. Phys. A 18 (1985), 2403; Phys. Rev. A 34 (1986), 2470; M. Jameel, J. Phys. A 19 (1986), 1967; B. Roy, Phys. Rev. A 34 (1986), 5108; A. Z. Tang and F. T. Chan, Phys. Rev. A 35 (1987), 911; R. Dutt and Y. P. Varshni, J. Phys. B 20 (1987), 2437; Y. P. Varshni, Phys. Rev. A 36 (1987), 3009; 38 (1988), 1595; 40 (1989), 2180; B. Roy and R. Roychoudhury, J. Phys. A 20 (1987), 3051; R. Roychoudhury and Y. P. Varshni, Phys. Rev. A 37 (1988), 2309; R. Sever and C. Tezcan, Phys. Rev. A 36 (1987), 1045; E. Papp, Phys. Rev. A 36 (1987), 3550; 38 (1988), 2158; H. Christiansen, L. N. Epele, H. Franchiotti, and C. A. Garcia Canal, Phys. Rev. A 40 (1989), 1760; S. M. Ikhdair and R. Sever, Z. Phys. D 28 (1993), 1.
62. 23. A. D. Dolgov and V. S. Popov, Phys. Lett. B 86 (1979), 185; H. J. de Vega, Commun. Math. Phys. 70 (1979), 29; A. V. Kudinov and M. A. Smondyrev, Czech. J. Phys. B 32 (1982), 556; Teor. Mat. Fiz. 56 (1983), 357 (Theor. Math. Phys. 56 (1983), 871); J. Avan and H. J. de Vega, Nucl. Phys. B 224 (1983), 61; Phys. Rev. D 29 (1984), 2891; 29 (1984), 2904; J. Avan, Nucl. Phys. B 237 (1984), 159; U. Sukhatme and T. Imbo, Phys. Rev. D 28 (1983), 418; R. S. Gangopadhyay, G. Ghosh, and B. Dutta-Roy, Phys. Rev. A 30 (1984), 594; A. Chatterjee, J. Phys. A 18 (1985), 2403; Phys. Rev. A 34 (1986), 2470; M. Jameel, J. Phys. A 19 (1986), 1967; B. Roy, Phys. Rev. A 34 (1986), 5108; A. Z. Tang and F. T. Chan, Phys. Rev. A 35 (1987), 911; R. Dutt and Y. P. Varshni, J. Phys. B 20 (1987), 2437; Y. P. Varshni, Phys. Rev. A 36 (1987), 3009; 38 (1988), 1595; 40 (1989), 2180; B. Roy and R. Roychoudhury, J. Phys. A 20 (1987), 3051; R. Roychoudhury and Y. P. Varshni, Phys. Rev. A 37 (1988), 2309; R. Sever and C. Tezcan, Phys. Rev. A 36 (1987), 1045; E. Papp, Phys. Rev. A 36 (1987), 3550; 38 (1988), 2158; H. Christiansen, L. N. Epele, H. Franchiotti, and C. A. Garcia Canal, Phys. Rev. A 40 (1989), 1760; S. M. Ikhdair and R. Sever, Z. Phys. D 28 (1993), 1.
63. 23. A. D. Dolgov and V. S. Popov, Phys. Lett. B 86 (1979), 185; H. J. de Vega, Commun. Math. Phys. 70 (1979), 29; A. V. Kudinov and M. A. Smondyrev, Czech. J. Phys. B 32 (1982), 556; Teor. Mat. Fiz. 56 (1983), 357 (Theor. Math. Phys. 56 (1983), 871); J. Avan and H. J. de Vega, Nucl. Phys. B 224 (1983), 61; Phys. Rev. D 29 (1984), 2891; 29 (1984), 2904; J. Avan, Nucl. Phys. B 237 (1984), 159; U. Sukhatme and T. Imbo, Phys. Rev. D 28 (1983), 418; R. S. Gangopadhyay, G. Ghosh, and B. Dutta-Roy, Phys. Rev. A 30 (1984), 594; A. Chatterjee, J. Phys. A 18 (1985), 2403; Phys. Rev. A 34 (1986), 2470; M. Jameel, J. Phys. A 19 (1986), 1967; B. Roy, Phys. Rev. A 34 (1986), 5108; A. Z. Tang and F. T. Chan, Phys. Rev. A 35 (1987), 911; R. Dutt and Y. P. Varshni, J. Phys. B 20 (1987), 2437; Y. P. Varshni, Phys. Rev. A 36 (1987), 3009; 38 (1988), 1595; 40 (1989), 2180; B. Roy and R. Roychoudhury, J. Phys. A 20 (1987), 3051; R. Roychoudhury and Y. P. Varshni, Phys. Rev. A 37 (1988), 2309; R. Sever and C. Tezcan, Phys. Rev. A 36 (1987), 1045; E. Papp, Phys. Rev. A 36 (1987), 3550; 38 (1988), 2158; H. Christiansen, L. N. Epele, H. Franchiotti, and C. A. Garcia Canal, Phys. Rev. A 40 (1989), 1760; S. M. Ikhdair and R. Sever, Z. Phys. D 28 (1993), 1.
64. 23. A. D. Dolgov and V. S. Popov, Phys. Lett. B 86 (1979), 185; H. J. de Vega, Commun. Math. Phys. 70 (1979), 29; A. V. Kudinov and M. A. Smondyrev, Czech. J. Phys. B 32 (1982), 556; Teor. Mat. Fiz. 56 (1983), 357 (Theor. Math. Phys. 56 (1983), 871); J. Avan and H. J. de Vega, Nucl. Phys. B 224 (1983), 61; Phys. Rev. D 29 (1984), 2891; 29 (1984), 2904; J. Avan, Nucl. Phys. B 237 (1984), 159; U. Sukhatme and T. Imbo, Phys. Rev. D 28 (1983), 418; R. S. Gangopadhyay, G. Ghosh, and B. Dutta-Roy, Phys. Rev. A 30 (1984), 594; A. Chatterjee, J. Phys. A 18 (1985), 2403; Phys. Rev. A 34 (1986), 2470; M. Jameel, J. Phys. A 19 (1986), 1967; B. Roy, Phys. Rev. A 34 (1986), 5108; A. Z. Tang and F. T. Chan, Phys. Rev. A 35 (1987), 911; R. Dutt and Y. P. Varshni, J. Phys. B 20 (1987), 2437; Y. P. Varshni, Phys. Rev. A 36 (1987), 3009; 38 (1988), 1595; 40 (1989), 2180; B. Roy and R. Roychoudhury, J. Phys. A 20 (1987), 3051; R. Roychoudhury and Y. P. Varshni, Phys. Rev. A 37 (1988), 2309; R. Sever and C. Tezcan, Phys. Rev. A 36 (1987), 1045; E. Papp, Phys. Rev. A 36 (1987), 3550; 38 (1988), 2158; H. Christiansen, L. N. Epele, H. Franchiotti, and C. A. Garcia Canal, Phys. Rev. A 40 (1989), 1760; S. M. Ikhdair and R. Sever, Z. Phys. D 28 (1993), 1.
65. 23. A. D. Dolgov and V. S. Popov, Phys. Lett. B 86 (1979), 185; H. J. de Vega, Commun. Math. Phys. 70 (1979), 29; A. V. Kudinov and M. A. Smondyrev, Czech. J. Phys. B 32 (1982), 556; Teor. Mat. Fiz. 56 (1983), 357 (Theor. Math. Phys. 56 (1983), 871); J. Avan and H. J. de Vega, Nucl. Phys. B 224 (1983), 61; Phys. Rev. D 29 (1984), 2891; 29 (1984), 2904; J. Avan, Nucl. Phys. B 237 (1984), 159; U. Sukhatme and T. Imbo, Phys. Rev. D 28 (1983), 418; R. S. Gangopadhyay, G. Ghosh, and B. Dutta-Roy, Phys. Rev. A 30 (1984), 594; A. Chatterjee, J. Phys. A 18 (1985), 2403; Phys. Rev. A 34 (1986), 2470; M. Jameel, J. Phys. A 19 (1986), 1967; B. Roy, Phys. Rev. A 34 (1986), 5108; A. Z. Tang and F. T. Chan, Phys. Rev. A 35 (1987), 911; R. Dutt and Y. P. Varshni, J. Phys. B 20 (1987), 2437; Y. P. Varshni, Phys. Rev. A 36 (1987), 3009; 38 (1988), 1595; 40 (1989), 2180; B. Roy and R. Roychoudhury, J. Phys. A 20 (1987), 3051; R. Roychoudhury and Y. P. Varshni, Phys. Rev. A 37 (1988), 2309; R. Sever and C. Tezcan, Phys. Rev. A 36 (1987), 1045; E. Papp, Phys. Rev. A 36 (1987), 3550; 38 (1988), 2158; H. Christiansen, L. N. Epele, H. Franchiotti, and C. A. Garcia Canal, Phys. Rev. A 40 (1989), 1760; S. M. Ikhdair and R. Sever, Z. Phys. D 28 (1993), 1.
66. 23. A. D. Dolgov and V. S. Popov, Phys. Lett. B 86 (1979), 185; H. J. de Vega, Commun. Math. Phys. 70 (1979), 29; A. V. Kudinov and M. A. Smondyrev, Czech. J. Phys. B 32 (1982), 556; Teor. Mat. Fiz. 56 (1983), 357 (Theor. Math. Phys. 56 (1983), 871); J. Avan and H. J. de Vega, Nucl. Phys. B 224 (1983), 61; Phys. Rev. D 29 (1984), 2891; 29 (1984), 2904; J. Avan, Nucl. Phys. B 237 (1984), 159; U. Sukhatme and T. Imbo, Phys. Rev. D 28 (1983), 418; R. S. Gangopadhyay, G. Ghosh, and B. Dutta-Roy, Phys. Rev. A 30 (1984), 594; A. Chatterjee, J. Phys. A 18 (1985), 2403; Phys. Rev. A 34 (1986), 2470; M. Jameel, J. Phys. A 19 (1986), 1967; B. Roy, Phys. Rev. A 34 (1986), 5108; A. Z. Tang and F. T. Chan, Phys. Rev. A 35 (1987), 911; R. Dutt and Y. P. Varshni, J. Phys. B 20 (1987), 2437; Y. P. Varshni, Phys. Rev. A 36 (1987), 3009; 38 (1988), 1595; 40 (1989), 2180; B. Roy and R. Roychoudhury, J. Phys. A 20 (1987), 3051; R. Roychoudhury and Y. P. Varshni, Phys. Rev. A 37 (1988), 2309; R. Sever and C. Tezcan, Phys. Rev. A 36 (1987), 1045; E. Papp, Phys. Rev. A 36 (1987), 3550; 38 (1988), 2158; H. Christiansen, L. N. Epele, H. Franchiotti, and C. A. Garcia Canal, Phys. Rev. A 40 (1989), 1760; S. M. Ikhdair and R. Sever, Z. Phys. D 28 (1993), 1.
67. 23. A. D. Dolgov and V. S. Popov, Phys. Lett. B 86 (1979), 185; H. J. de Vega, Commun. Math. Phys. 70 (1979), 29; A. V. Kudinov and M. A. Smondyrev, Czech. J. Phys. B 32 (1982), 556; Teor. Mat. Fiz. 56 (1983), 357 (Theor. Math. Phys. 56 (1983), 871); J. Avan and H. J. de Vega, Nucl. Phys. B 224 (1983), 61; Phys. Rev. D 29 (1984), 2891; 29 (1984), 2904; J. Avan, Nucl. Phys. B 237 (1984), 159; U. Sukhatme and T. Imbo, Phys. Rev. D 28 (1983), 418; R. S. Gangopadhyay, G. Ghosh, and B. Dutta-Roy, Phys. Rev. A 30 (1984), 594; A. Chatterjee, J. Phys. A 18 (1985), 2403; Phys. Rev. A 34 (1986), 2470; M. Jameel, J. Phys. A 19 (1986), 1967; B. Roy, Phys. Rev. A 34 (1986), 5108; A. Z. Tang and F. T. Chan, Phys. Rev. A 35 (1987), 911; R. Dutt and Y. P. Varshni, J. Phys. B 20 (1987), 2437; Y. P. Varshni, Phys. Rev. A 36 (1987), 3009; 38 (1988), 1595; 40 (1989), 2180; B. Roy and R. Roychoudhury, J. Phys. A 20 (1987), 3051; R. Roychoudhury and Y. P. Varshni, Phys. Rev. A 37 (1988), 2309; R. Sever and C. Tezcan, Phys. Rev. A 36 (1987), 1045; E. Papp, Phys. Rev. A 36 (1987), 3550; 38 (1988), 2158; H. Christiansen, L. N. Epele, H. Franchiotti, and C. A. Garcia Canal, Phys. Rev. A 40 (1989), 1760; S. M. Ikhdair and R. Sever, Z. Phys. D 28 (1993), 1.
68. 23. A. D. Dolgov and V. S. Popov, Phys. Lett. B 86 (1979), 185; H. J. de Vega, Commun. Math. Phys. 70 (1979), 29; A. V. Kudinov and M. A. Smondyrev, Czech. J. Phys. B 32 (1982), 556; Teor. Mat. Fiz. 56 (1983), 357 (Theor. Math. Phys. 56 (1983), 871); J. Avan and H. J. de Vega, Nucl. Phys. B 224 (1983), 61; Phys. Rev. D 29 (1984), 2891; 29 (1984), 2904; J. Avan, Nucl. Phys. B 237 (1984), 159; U. Sukhatme and T. Imbo, Phys. Rev. D 28 (1983), 418; R. S. Gangopadhyay, G. Ghosh, and B. Dutta-Roy, Phys. Rev. A 30 (1984), 594; A. Chatterjee, J. Phys. A 18 (1985), 2403; Phys. Rev. A 34 (1986), 2470; M. Jameel, J. Phys. A 19 (1986), 1967; B. Roy, Phys. Rev. A 34 (1986), 5108; A. Z. Tang and F. T. Chan, Phys. Rev. A 35 (1987), 911; R. Dutt and Y. P. Varshni, J. Phys. B 20 (1987), 2437; Y. P. Varshni, Phys. Rev. A 36 (1987), 3009; 38 (1988), 1595; 40 (1989), 2180; B. Roy and R. Roychoudhury, J. Phys. A 20 (1987), 3051; R. Roychoudhury and Y. P. Varshni, Phys. Rev. A 37 (1988), 2309; R. Sever and C. Tezcan, Phys. Rev. A 36 (1987), 1045; E. Papp, Phys. Rev. A 36 (1987), 3550; 38 (1988), 2158; H. Christiansen, L. N. Epele, H. Franchiotti, and C. A. Garcia Canal, Phys. Rev. A 40 (1989), 1760; S. M. Ikhdair and R. Sever, Z. Phys. D 28 (1993), 1.
69. 23. A. D. Dolgov and V. S. Popov, Phys. Lett. B 86 (1979), 185; H. J. de Vega, Commun. Math. Phys. 70 (1979), 29; A. V. Kudinov and M. A. Smondyrev, Czech. J. Phys. B 32 (1982), 556; Teor. Mat. Fiz. 56 (1983), 357 (Theor. Math. Phys. 56 (1983), 871); J. Avan and H. J. de Vega, Nucl. Phys. B 224 (1983), 61; Phys. Rev. D 29 (1984), 2891; 29 (1984), 2904; J. Avan, Nucl. Phys. B 237 (1984), 159; U. Sukhatme and T. Imbo, Phys. Rev. D 28 (1983), 418; R. S. Gangopadhyay, G. Ghosh, and B. Dutta-Roy, Phys. Rev. A 30 (1984), 594; A. Chatterjee, J. Phys. A 18 (1985), 2403; Phys. Rev. A 34 (1986), 2470; M. Jameel, J. Phys. A 19 (1986), 1967; B. Roy, Phys. Rev. A 34 (1986), 5108; A. Z. Tang and F. T. Chan, Phys. Rev. A 35 (1987), 911; R. Dutt and Y. P. Varshni, J. Phys. B 20 (1987), 2437; Y. P. Varshni, Phys. Rev. A 36 (1987), 3009; 38 (1988), 1595; 40 (1989), 2180; B. Roy and R. Roychoudhury, J. Phys. A 20 (1987), 3051; R. Roychoudhury and Y. P. Varshni, Phys. Rev. A 37 (1988), 2309; R. Sever and C. Tezcan, Phys. Rev. A 36 (1987), 1045; E. Papp, Phys. Rev. A 36 (1987), 3550; 38 (1988), 2158; H. Christiansen, L. N. Epele, H. Franchiotti, and C. A. Garcia Canal, Phys. Rev. A 40 (1989), 1760; S. M. Ikhdair and R. Sever, Z. Phys. D 28 (1993), 1.
70. 23. A. D. Dolgov and V. S. Popov, Phys. Lett. B 86 (1979), 185; H. J. de Vega, Commun. Math. Phys. 70 (1979), 29; A. V. Kudinov and M. A. Smondyrev, Czech. J. Phys. B 32 (1982), 556; Teor. Mat. Fiz. 56 (1983), 357 (Theor. Math. Phys. 56 (1983), 871); J. Avan and H. J. de Vega, Nucl. Phys. B 224 (1983), 61; Phys. Rev. D 29 (1984), 2891; 29 (1984), 2904; J. Avan, Nucl. Phys. B 237 (1984), 159; U. Sukhatme and T. Imbo, Phys. Rev. D 28 (1983), 418; R. S. Gangopadhyay, G. Ghosh, and B. Dutta-Roy, Phys. Rev. A 30 (1984), 594; A. Chatterjee, J. Phys. A 18 (1985), 2403; Phys. Rev. A 34 (1986), 2470; M. Jameel, J. Phys. A 19 (1986), 1967; B. Roy, Phys. Rev. A 34 (1986), 5108; A. Z. Tang and F. T. Chan, Phys. Rev. A 35 (1987), 911; R. Dutt and Y. P. Varshni, J. Phys. B 20 (1987), 2437; Y. P. Varshni, Phys. Rev. A 36 (1987), 3009; 38 (1988), 1595; 40 (1989), 2180; B. Roy and R. Roychoudhury, J. Phys. A 20 (1987), 3051; R. Roychoudhury and Y. P. Varshni, Phys. Rev. A 37 (1988), 2309; R. Sever and C. Tezcan, Phys. Rev. A 36 (1987), 1045; E. Papp, Phys. Rev. A 36 (1987), 3550; 38 (1988), 2158; H. Christiansen, L. N. Epele, H. Franchiotti, and C. A. Garcia Canal, Phys. Rev. A 40 (1989), 1760; S. M. Ikhdair and R. Sever, Z. Phys. D 28 (1993), 1.
71. 23. A. D. Dolgov and V. S. Popov, Phys. Lett. B 86 (1979), 185; H. J. de Vega, Commun. Math. Phys. 70 (1979), 29; A. V. Kudinov and M. A. Smondyrev, Czech. J. Phys. B 32 (1982), 556; Teor. Mat. Fiz. 56 (1983), 357 (Theor. Math. Phys. 56 (1983), 871); J. Avan and H. J. de Vega, Nucl. Phys. B 224 (1983), 61; Phys. Rev. D 29 (1984), 2891; 29 (1984), 2904; J. Avan, Nucl. Phys. B 237 (1984), 159; U. Sukhatme and T. Imbo, Phys. Rev. D 28 (1983), 418; R. S. Gangopadhyay, G. Ghosh, and B. Dutta-Roy, Phys. Rev. A 30 (1984), 594; A. Chatterjee, J. Phys. A 18 (1985), 2403; Phys. Rev. A 34 (1986), 2470; M. Jameel, J. Phys. A 19 (1986), 1967; B. Roy, Phys. Rev. A 34 (1986), 5108; A. Z. Tang and F. T. Chan, Phys. Rev. A 35 (1987), 911; R. Dutt and Y. P. Varshni, J. Phys. B 20 (1987), 2437; Y. P. Varshni, Phys. Rev. A 36 (1987), 3009; 38 (1988), 1595; 40 (1989), 2180; B. Roy and R. Roychoudhury, J. Phys. A 20 (1987), 3051; R. Roychoudhury and Y. P. Varshni, Phys. Rev. A 37 (1988), 2309; R. Sever and C. Tezcan, Phys. Rev. A 36 (1987), 1045; E. Papp, Phys. Rev. A 36 (1987), 3550; 38 (1988), 2158; H. Christiansen, L. N. Epele, H. Franchiotti, and C. A. Garcia Canal, Phys. Rev. A 40 (1989), 1760; S. M. Ikhdair and R. Sever, Z. Phys. D 28 (1993), 1.
72. 23. A. D. Dolgov and V. S. Popov, Phys. Lett. B 86 (1979), 185; H. J. de Vega, Commun. Math. Phys. 70 (1979), 29; A. V. Kudinov and M. A. Smondyrev, Czech. J. Phys. B 32 (1982), 556; Teor. Mat. Fiz. 56 (1983), 357 (Theor. Math. Phys. 56 (1983), 871); J. Avan and H. J. de Vega, Nucl. Phys. B 224 (1983), 61; Phys. Rev. D 29 (1984), 2891; 29 (1984), 2904; J. Avan, Nucl. Phys. B 237 (1984), 159; U. Sukhatme and T. Imbo, Phys. Rev. D 28 (1983), 418; R. S. Gangopadhyay, G. Ghosh, and B. Dutta-Roy, Phys. Rev. A 30 (1984), 594; A. Chatterjee, J. Phys. A 18 (1985), 2403; Phys. Rev. A 34 (1986), 2470; M. Jameel, J. Phys. A 19 (1986), 1967; B. Roy, Phys. Rev. A 34 (1986), 5108; A. Z. Tang and F. T. Chan, Phys. Rev. A 35 (1987), 911; R. Dutt and Y. P. Varshni, J. Phys. B 20 (1987), 2437; Y. P. Varshni, Phys. Rev. A 36 (1987), 3009; 38 (1988), 1595; 40 (1989), 2180; B. Roy and R. Roychoudhury, J. Phys. A 20 (1987), 3051; R. Roychoudhury and Y. P. Varshni, Phys. Rev. A 37 (1988), 2309; R. Sever and C. Tezcan, Phys. Rev. A 36 (1987), 1045; E. Papp, Phys. Rev. A 36 (1987), 3550; 38 (1988), 2158; H. Christiansen, L. N. Epele, H. Franchiotti, and C. A. Garcia Canal, Phys. Rev. A 40 (1989), 1760; S. M. Ikhdair and R. Sever, Z. Phys. D 28 (1993), 1.
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77. 26. S. Atag, Phys. Rev. A 37 (1988), 2280; D. A. Morales, Chem. Phys. Lett. 161 (1989), 253; M. Bag and R. Dutt, J. Phys. B 23 (1990), 3075; Y. P. Varshni, Can. J. Phys. 71 (1993), 122; Chem. Phys. 188 (1994), 197.
78. 26. S. Atag, Phys. Rev. A 37 (1988), 2280; D. A. Morales, Chem. Phys. Lett. 161 (1989), 253; M. Bag and R. Dutt, J. Phys. B 23 (1990), 3075; Y. P. Varshni, Can. J. Phys. 71 (1993), 122; Chem. Phys. 188 (1994), 197.
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81. 28. S. Hikami and E. Brézin, J. Phys. A 12 (1979), 759; T. Imbo, A. Pagnamenta, and U. Sukhatme, Phys. Lett. A 105 (1984), 183; G. Moreno and A. Zepeda, J. Phys. B 17 (1984), 21; A. Chatterjee, J. Phys. A 18 (1985), 1193; 19 (1986), 3707; Phys. Rev. A 35 (1987), 2722; Ş. Erkoç and R. Sever, Phys. Rev. A 33 (1986), 588; C. H. Lai, J. Math. Phys. 28 (1987), 1801; R. Dutt, U. Mukherji, and Y. P. Varshni, Phys. Rev. A 34 (1986), 777; J. Phys. B 19 (1986), 3411; R. Dutt and Y. P. Varshni, Z. Phys. D 2 (1986), 207; Y. P. Varshni, Phys. Rev. A 36 (1987), 3009; R. Sever and C. Tezcan, Phys. Rev. A 35 (1987), 2725; 37 (1988), 3158; K. Andrew and J. Supplee, Am. J. Phys. 58 (1990), 1177; P. K. Bera, S. Bhattacharyya, U. Das, and B. Talukdar, Phys. Rev. A 48 (1993), 4764.
82. 28. S. Hikami and E. Brézin, J. Phys. A 12 (1979), 759; T. Imbo, A. Pagnamenta, and U. Sukhatme, Phys. Lett. A 105 (1984), 183; G. Moreno and A. Zepeda, J. Phys. B 17 (1984), 21; A. Chatterjee, J. Phys. A 18 (1985), 1193; 19 (1986), 3707; Phys. Rev. A 35 (1987), 2722; Ş. Erkoç and R. Sever, Phys. Rev. A 33 (1986), 588; C. H. Lai, J. Math. Phys. 28 (1987), 1801; R. Dutt, U. Mukherji, and Y. P. Varshni, Phys. Rev. A 34 (1986), 777; J. Phys. B 19 (1986), 3411; R. Dutt and Y. P. Varshni, Z. Phys. D 2 (1986), 207; Y. P. Varshni, Phys. Rev. A 36 (1987), 3009; R. Sever and C. Tezcan, Phys. Rev. A 35 (1987), 2725; 37 (1988), 3158; K. Andrew and J. Supplee, Am. J. Phys. 58 (1990), 1177; P. K. Bera, S. Bhattacharyya, U. Das, and B. Talukdar, Phys. Rev. A 48 (1993), 4764.
83. 28. S. Hikami and E. Brézin, J. Phys. A 12 (1979), 759; T. Imbo, A. Pagnamenta, and U. Sukhatme, Phys. Lett. A 105 (1984), 183; G. Moreno and A. Zepeda, J. Phys. B 17 (1984), 21; A. Chatterjee, J. Phys. A 18 (1985), 1193; 19 (1986), 3707; Phys. Rev. A 35 (1987), 2722; Ş. Erkoç and R. Sever, Phys. Rev. A 33 (1986), 588; C. H. Lai, J. Math. Phys. 28 (1987), 1801; R. Dutt, U. Mukherji, and Y. P. Varshni, Phys. Rev. A 34 (1986), 777; J. Phys. B 19 (1986), 3411; R. Dutt and Y. P. Varshni, Z. Phys. D 2 (1986), 207; Y. P. Varshni, Phys. Rev. A 36 (1987), 3009; R. Sever and C. Tezcan, Phys. Rev. A 35 (1987), 2725; 37 (1988), 3158; K. Andrew and J. Supplee, Am. J. Phys. 58 (1990), 1177; P. K. Bera, S. Bhattacharyya, U. Das, and B. Talukdar, Phys. Rev. A 48 (1993), 4764.
84. 28. S. Hikami and E. Brézin, J. Phys. A 12 (1979), 759; T. Imbo, A. Pagnamenta, and U. Sukhatme, Phys. Lett. A 105 (1984), 183; G. Moreno and A. Zepeda, J. Phys. B 17 (1984), 21; A. Chatterjee, J. Phys. A 18 (1985), 1193; 19 (1986), 3707; Phys. Rev. A 35 (1987), 2722; Ş. Erkoç and R. Sever, Phys. Rev. A 33 (1986), 588; C. H. Lai, J. Math. Phys. 28 (1987), 1801; R. Dutt, U. Mukherji, and Y. P. Varshni, Phys. Rev. A 34 (1986), 777; J. Phys. B 19 (1986), 3411; R. Dutt and Y. P. Varshni, Z. Phys. D 2 (1986), 207; Y. P. Varshni, Phys. Rev. A 36 (1987), 3009; R. Sever and C. Tezcan, Phys. Rev. A 35 (1987), 2725; 37 (1988), 3158; K. Andrew and J. Supplee, Am. J. Phys. 58 (1990), 1177; P. K. Bera, S. Bhattacharyya, U. Das, and B. Talukdar, Phys. Rev. A 48 (1993), 4764.
85. 28. S. Hikami and E. Brézin, J. Phys. A 12 (1979), 759; T. Imbo, A. Pagnamenta, and U. Sukhatme, Phys. Lett. A 105 (1984), 183; G. Moreno and A. Zepeda, J. Phys. B 17 (1984), 21; A. Chatterjee, J. Phys. A 18 (1985), 1193; 19 (1986), 3707; Phys. Rev. A 35 (1987), 2722; Ş. Erkoç and R. Sever, Phys. Rev. A 33 (1986), 588; C. H. Lai, J. Math. Phys. 28 (1987), 1801; R. Dutt, U. Mukherji, and Y. P. Varshni, Phys. Rev. A 34 (1986), 777; J. Phys. B 19 (1986), 3411; R. Dutt and Y. P. Varshni, Z. Phys. D 2 (1986), 207; Y. P. Varshni, Phys. Rev. A 36 (1987), 3009; R. Sever and C. Tezcan, Phys. Rev. A 35 (1987), 2725; 37 (1988), 3158; K. Andrew and J. Supplee, Am. J. Phys. 58 (1990), 1177; P. K. Bera, S. Bhattacharyya, U. Das, and B. Talukdar, Phys. Rev. A 48 (1993), 4764.
86. 28. S. Hikami and E. Brézin, J. Phys. A 12 (1979), 759; T. Imbo, A. Pagnamenta, and U. Sukhatme, Phys. Lett. A 105 (1984), 183; G. Moreno and A. Zepeda, J. Phys. B 17 (1984), 21; A. Chatterjee, J. Phys. A 18 (1985), 1193; 19 (1986), 3707; Phys. Rev. A 35 (1987), 2722; Ş. Erkoç and R. Sever, Phys. Rev. A 33 (1986), 588; C. H. Lai, J. Math. Phys. 28 (1987), 1801; R. Dutt, U. Mukherji, and Y. P. Varshni, Phys. Rev. A 34 (1986), 777; J. Phys. B 19 (1986), 3411; R. Dutt and Y. P. Varshni, Z. Phys. D 2 (1986), 207; Y. P. Varshni, Phys. Rev. A 36 (1987), 3009; R. Sever and C. Tezcan, Phys. Rev. A 35 (1987), 2725; 37 (1988), 3158; K. Andrew and J. Supplee, Am. J. Phys. 58 (1990), 1177; P. K. Bera, S. Bhattacharyya, U. Das, and B. Talukdar, Phys. Rev. A 48 (1993), 4764.
87. 28. S. Hikami and E. Brézin, J. Phys. A 12 (1979), 759; T. Imbo, A. Pagnamenta, and U. Sukhatme, Phys. Lett. A 105 (1984), 183; G. Moreno and A. Zepeda, J. Phys. B 17 (1984), 21; A. Chatterjee, J. Phys. A 18 (1985), 1193; 19 (1986), 3707; Phys. Rev. A 35 (1987), 2722; Ş. Erkoç and R. Sever, Phys. Rev. A 33 (1986), 588; C. H. Lai, J. Math. Phys. 28 (1987), 1801; R. Dutt, U. Mukherji, and Y. P. Varshni, Phys. Rev. A 34 (1986), 777; J. Phys. B 19 (1986), 3411; R. Dutt and Y. P. Varshni, Z. Phys. D 2 (1986), 207; Y. P. Varshni, Phys. Rev. A 36 (1987), 3009; R. Sever and C. Tezcan, Phys. Rev. A 35 (1987), 2725; 37 (1988), 3158; K. Andrew and J. Supplee, Am. J. Phys. 58 (1990), 1177; P. K. Bera, S. Bhattacharyya, U. Das, and B. Talukdar, Phys. Rev. A 48 (1993), 4764.
88. 28. S. Hikami and E. Brézin, J. Phys. A 12 (1979), 759; T. Imbo, A. Pagnamenta, and U. Sukhatme, Phys. Lett. A 105 (1984), 183; G. Moreno and A. Zepeda, J. Phys. B 17 (1984), 21; A. Chatterjee, J. Phys. A 18 (1985), 1193; 19 (1986), 3707; Phys. Rev. A 35 (1987), 2722; Ş. Erkoç and R. Sever, Phys. Rev. A 33 (1986), 588; C. H. Lai, J. Math. Phys. 28 (1987), 1801; R. Dutt, U. Mukherji, and Y. P. Varshni, Phys. Rev. A 34 (1986), 777; J. Phys. B 19 (1986), 3411; R. Dutt and Y. P. Varshni, Z. Phys. D 2 (1986), 207; Y. P. Varshni, Phys. Rev. A 36 (1987), 3009; R. Sever and C. Tezcan, Phys. Rev. A 35 (1987), 2725; 37 (1988), 3158; K. Andrew and J. Supplee, Am. J. Phys. 58 (1990), 1177; P. K. Bera, S. Bhattacharyya, U. Das, and B. Talukdar, Phys. Rev. A 48 (1993), 4764.
89. 28. S. Hikami and E. Brézin, J. Phys. A 12 (1979), 759; T. Imbo, A. Pagnamenta, and U. Sukhatme, Phys. Lett. A 105 (1984), 183; G. Moreno and A. Zepeda, J. Phys. B 17 (1984), 21; A. Chatterjee, J. Phys. A 18 (1985), 1193; 19 (1986), 3707; Phys. Rev. A 35 (1987), 2722; Ş. Erkoç and R. Sever, Phys. Rev. A 33 (1986), 588; C. H. Lai, J. Math. Phys. 28 (1987), 1801; R. Dutt, U. Mukherji, and Y. P. Varshni, Phys. Rev. A 34 (1986), 777; J. Phys. B 19 (1986), 3411; R. Dutt and Y. P. Varshni, Z. Phys. D 2 (1986), 207; Y. P. Varshni, Phys. Rev. A 36 (1987), 3009; R. Sever and C. Tezcan, Phys. Rev. A 35 (1987), 2725; 37 (1988), 3158; K. Andrew and J. Supplee, Am. J. Phys. 58 (1990), 1177; P. K. Bera, S. Bhattacharyya, U. Das, and B. Talukdar, Phys. Rev. A 48 (1993), 4764.
90. 28. S. Hikami and E. Brézin, J. Phys. A 12 (1979), 759; T. Imbo, A. Pagnamenta, and U. Sukhatme, Phys. Lett. A 105 (1984), 183; G. Moreno and A. Zepeda, J. Phys. B 17 (1984), 21; A. Chatterjee, J. Phys. A 18 (1985), 1193; 19 (1986), 3707; Phys. Rev. A 35 (1987), 2722; Ş. Erkoç and R. Sever, Phys. Rev. A 33 (1986), 588; C. H. Lai, J. Math. Phys. 28 (1987), 1801; R. Dutt, U. Mukherji, and Y. P. Varshni, Phys. Rev. A 34 (1986), 777; J. Phys. B 19 (1986), 3411; R. Dutt and Y. P. Varshni, Z. Phys. D 2 (1986), 207; Y. P. Varshni, Phys. Rev. A 36 (1987), 3009; R. Sever and C. Tezcan, Phys. Rev. A 35 (1987), 2725; 37 (1988), 3158; K. Andrew and J. Supplee, Am. J. Phys. 58 (1990), 1177; P. K. Bera, S. Bhattacharyya, U. Das, and B. Talukdar, Phys. Rev. A 48 (1993), 4764.
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298. 129. G. S. Ezra, K. Richter, G. Tanner, and D. Wintgen, J. Phys. B 24 (1991), L413; J. Müller, J. Burgdörfer, and D. Noid, Phys. Rev. A 45 (1992), 1471; D. Wintgen, K. Richter, and G. Tanner, CHAOS 2 (1992), 19; in "Quantum Chaos" (G. Casati, I. Guarneri, and U. Smilansky, Eds.), Proceedings of the International School of Physics "Enrico Fermi", Course 119, p. 113, North-Holland, Amsterdam, 1993; D. Wintgen, A. Bürgers, K. Richter, and G. Tanner, Prog. Theor. Phys. Suppl. 116 (1994), 121; P. V. Grujić and N. S. Simonović, J. Phys. B 28 (1995), 1159; and the references therein.
299. 129. G. S. Ezra, K. Richter, G. Tanner, and D. Wintgen, J. Phys. B 24 (1991), L413; J. Müller, J. Burgdörfer, and D. Noid, Phys. Rev. A 45 (1992), 1471; D. Wintgen, K. Richter, and G. Tanner, CHAOS 2 (1992), 19; in "Quantum Chaos" (G. Casati, I. Guarneri, and U. Smilansky, Eds.), Proceedings of the International School of Physics "Enrico Fermi", Course 119, p. 113, North-Holland, Amsterdam, 1993; D. Wintgen, A. Bürgers, K. Richter, and G. Tanner, Prog. Theor. Phys. Suppl. 116 (1994), 121; P. V. Grujić and N. S. Simonović, J. Phys. B 28 (1995), 1159; and the references therein.
300. 129. G. S. Ezra, K. Richter, G. Tanner, and D. Wintgen, J. Phys. B 24 (1991), L413; J. Müller, J. Burgdörfer, and D. Noid, Phys. Rev. A 45 (1992), 1471; D. Wintgen, K. Richter, and G. Tanner, CHAOS 2 (1992), 19; in "Quantum Chaos" (G. Casati, I. Guarneri, and U. Smilansky, Eds.), Proceedings of the International School of Physics "Enrico Fermi", Course 119, p. 113, North-Holland, Amsterdam, 1993; D. Wintgen, A. Bürgers, K. Richter, and G. Tanner, Prog. Theor. Phys. Suppl. 116 (1994), 121; P. V. Grujić and N. S. Simonović, J. Phys. B 28 (1995), 1159; and the references therein.
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303. 130. P. G. Harris, H. C. Bryant, A. H. Mohagheghi, R. A. Reeder, H. Sharifian, H. Tootoonchi, C. Y. Tang, J. B. Donahue, C. R. Quick, D. C. Rislove, and W. W. Smith, Phys. Rev. Lett. 65 (1990), 309; P. G. Harris, H. C. Bryant, A. H. Mohagheghi, R. A. Reeder, C. Y. Tang, J. B. Donahue, and C. R. Quick, Phys. Rev. A 42 (1990), 6443; M. Domke, C. Xue, A. Puschmann, T. Mandel, E. Hudson, D. A. Shirley, G. Kaindl, C. H. Greene, and H. R. Sadeghpour, Phys. Rev. Lett. 66 (1991), 1306; M. Domke, G. Remmers, and G. Kaindl, Phys. Rev. Lett. 69 (1992), 1171; M. Domke, K. Schulz, G. Remmer, A. Gutiérrez, G. Kaindl, and D. Wintgen, Phys. Rev. A 51 (1995), R4309.
304. 130. P. G. Harris, H. C. Bryant, A. H. Mohagheghi, R. A. Reeder, H. Sharifian, H. Tootoonchi, C. Y. Tang, J. B. Donahue, C. R. Quick, D. C. Rislove, and W. W. Smith, Phys. Rev. Lett. 65 (1990), 309; P. G. Harris, H. C. Bryant, A. H. Mohagheghi, R. A. Reeder, C. Y. Tang, J. B. Donahue, and C. R. Quick, Phys. Rev. A 42 (1990), 6443; M. Domke, C. Xue, A. Puschmann, T. Mandel, E. Hudson, D. A. Shirley, G. Kaindl, C. H. Greene, and H. R. Sadeghpour, Phys. Rev. Lett. 66 (1991), 1306; M. Domke, G. Remmers, and G. Kaindl, Phys. Rev. Lett. 69 (1992), 1171; M. Domke, K. Schulz, G. Remmer, A. Gutiérrez, G. Kaindl, and D. Wintgen, Phys. Rev. A 51 (1995), R4309.
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306. 131. C. H. Townes and A. L. Schawlow, "Microwave Spectrocopy," p. 300, McGraw-Hill, New York, 1955; for other examples in molecular, condensed matter, nuclear and particle physics see Ref. [101].
307. 132. T. D. Holstein, J. Chem. Phys. 56 (1952), 832; Westinghouse Research Report 60-94698-3-R9, 1955; C. Herring, Rev. Mod. Phys. 34 (1962), 631; L. D. Landau and E. M. Lifshitz, "Quantum Mechanics," 2nd ed., p. 312, Academic Press, New York, 1965.
308. 132. T. D. Holstein, J. Chem. Phys. 56 (1952), 832; Westinghouse Research Report 60-94698-3-R9, 1955; C. Herring, Rev. Mod. Phys. 34 (1962), 631; L. D. Landau and E. M. Lifshitz, "Quantum Mechanics," 2nd ed., p. 312, Academic Press, New York, 1965.
309. 132. T. D. Holstein, J. Chem. Phys. 56 (1952), 832; Westinghouse Research Report 60-94698-3-R9, 1955; C. Herring, Rev. Mod. Phys. 34 (1962), 631; L. D. Landau and E. M. Lifshitz, "Quantum Mechanics," 2nd ed., p. 312, Academic Press, New York, 1965.
310. 132. T. D. Holstein, J. Chem. Phys. 56 (1952), 832; Westinghouse Research Report 60-94698-3-R9, 1955; C. Herring, Rev. Mod. Phys. 34 (1962), 631; L. D. Landau and E. M. Lifshitz, "Quantum Mechanics," 2nd ed., p. 312, Academic Press, New York, 1965.
311. 133. C. D. Lin, Phys. Rev. A 29 (1984), 1019.
312. 134. M. Dunn and D.K. Watson, Continuation of the Schrödinger Equation for higher angular momentum states of two-electron atoms to D dimensions and interdimensional degeneracies, Few-Body Systems, accepted.
313. 135. M. H. Padé, Ann. Sci. Ecole Norm. Sup. 9, Suppl. (1892), 1; R. E. Shafer, SIAM J. Numer. Anal. 11 (1972), 447.
314. 135. M. H. Padé, Ann. Sci. Ecole Norm. Sup. 9, Suppl. (1892), 1; R. E. Shafer, SIAM J. Numer. Anal. 11 (1972), 447.
315. 136. Reference [53, Appendix A].
316. 137. M. Dunn and D.K. Watson, "The Large Dimension Limit of Higher Angular Momentum States," to be submitted.
317. 138. J. D. Talman, "Special Functions: A Group Theoretic Approach," W. A. Benjamin, New York, 1968; especially Section 7-2, which discusses the completeness of the script Ds.
318. 139. B. G. Wybourne, "Classical Groups for Physicists," Chap. 15, Wiley, New York, 1974.
319. 140. For some states in even dimensions the sum of Eq. (1) will involve a sum over two different irreducible representations since irreducible representations of O(D) with self-associate Young tableaux reduce to two inequivalent representations of SO(D) of the same dimensionality [141].
320. 141. References [142, pp. 155, 164; 143, p. 396].
321. 142. H. Weyl, "The Classical Groups," Princeton Univ. Press, Princeton, NJ, 1939.
322. 143. M. Hamermesh, "Group Theory and Its Application to Physical Problems," Dover, New York, 1989; Chap. 10 is an elementary introduction to the method of irreducible tensors.
323. 144. Wybourne, op. cit., Chap. 10.
324. 145. A simplified, heuristic account of this work for the two electron problem by the present authors may be found in [8, p. 375].
325. 146. C. Schwartz, Phys. Rev. 123 (1961), 1700.
326. 147. See also C. Bottcher, D. R. Schultz, and D. H. Madison, Phys. Rev. A 49 (1994), 1714; D. R. Schultz, C. Bottcher, D. H. Madison, J. L. Peacher, G. Buffington, M. S. Pindzola, T. W. Gorczyca, P. Gavras, and D. C. Griffin, Phys. Rev. A 50 (1994), 1348.
327. 147. See also C. Bottcher, D. R. Schultz, and D. H. Madison, Phys. Rev. A 49 (1994), 1714; D. R. Schultz, C. Bottcher, D. H. Madison, J. L. Peacher, G. Buffington, M. S. Pindzola, T. W. Gorczyca, P. Gavras, and D. C. Griffin, Phys. Rev. A 50 (1994), 1348.
328. 148. We follow the original approach to irreducible tensors mainly due to Weyl [142, 143, 149, 150], although in this paper the expression of this method through primitive invariant projection operators is somewhat different. An approach based on the concept of induced representations has been developed by Godement and Zhelobenko, see for example Ref. [151]. A direct approach to finding the primitive invariant projection operators of O(D) using the primitive invariant tensors has been developed by Cvitanović and others; see for example Ref. [152] (also see Ref. [178]).
329. 149. H. Boerner, "Representations of Groups," North-Holland, Amsterdam, 1963. Although it does not explicitly discuss the application of the method of irreducible tensors to O(D) the discussion of the general principles is quite accessible.
330. 150. C. D. H. Chisholm, "Group Theoretical Techniques in Quantum Chemistry," Academic Press, London, 1976; Chap. 8 is an elementary introduction.
331. 151. A. O. Barut and R. Raczka, "Theory of Group Representations and Applications," p. 291, PWN, Warsaw, 1977.
332. 152. P. Cvitanović, "Group Theory," part I, Nordita, Copenhagen, 1984.
333. 153. M. Dunn and D. K. Watson, Continuation of the wave function for higher angular momentum states to D dimensions. II. Elimination of linear dependencies, Ann. Phys. (N.Y.) 251 (1996), 319.
334. 154. Young diagrams go by a variety of names in the literature. We follow the nomenclature of G. de B. Robinson, "Representation Theory of the Symmetric Group," University of Toronto Press, Toronto, 1961; A. Messiah, "Quantum Mechanics," Vol. II, North Holland, Amsterdam, 1966; and M. Weissbluth, "Atoms and Molecules," Academic Press, New York, 1978. They also go by the name 'pattern' in Ref. [155]; by the names 'graph', 'diagram', and 'Young Pattern' in Ref. [143]; by the name 'shape' in D. E. Rutherford, "Substitutional Analysis," Edinburgh University Press, Edinburgh, 1948 and C. D. H. Chisholm, op. cit., and by 'frame' in Ref. [149]. Reference [142] reserves the name 'diagram' for Young diagrams and one of the standard Young tableaux. In D. B. Lichtenberg, "Unitary Symmetry and Elementary Particles," 1st ed., Academic Press, New York, 1970, Young diagrams are referred to as either 'Young diagrams' or 'Young tableaux'.
335. 154. Young diagrams go by a variety of names in the literature. We follow the nomenclature of G. de B. Robinson, "Representation Theory of the Symmetric Group," University of Toronto Press, Toronto, 1961; A. Messiah, "Quantum Mechanics," Vol. II, North Holland, Amsterdam, 1966; and M. Weissbluth, "Atoms and Molecules," Academic Press, New York, 1978. They also go by the name 'pattern' in Ref. [155]; by the names 'graph', 'diagram', and 'Young Pattern' in Ref. [143]; by the name 'shape' in D. E. Rutherford, "Substitutional Analysis," Edinburgh University Press, Edinburgh, 1948 and C. D. H. Chisholm, op. cit., and by 'frame' in Ref. [149]. Reference [142] reserves the name 'diagram' for Young diagrams and one of the standard Young tableaux. In D. B. Lichtenberg, "Unitary Symmetry and Elementary Particles," 1st ed., Academic Press, New York, 1970, Young diagrams are referred to as either 'Young diagrams' or 'Young tableaux'.
336. 154. Young diagrams go by a variety of names in the literature. We follow the nomenclature of G. de B. Robinson, "Representation Theory of the Symmetric Group," University of Toronto Press, Toronto, 1961; A. Messiah, "Quantum Mechanics," Vol. II, North Holland, Amsterdam, 1966; and M. Weissbluth, "Atoms and Molecules," Academic Press, New York, 1978. They also go by the name 'pattern' in Ref. [155]; by the names 'graph', 'diagram', and 'Young Pattern' in Ref. [143]; by the name 'shape' in D. E. Rutherford, "Substitutional Analysis," Edinburgh University Press, Edinburgh, 1948 and C. D. H. Chisholm, op. cit., and by 'frame' in Ref. [149]. Reference [142] reserves the name 'diagram' for Young diagrams and one of the standard Young tableaux. In D. B. Lichtenberg, "Unitary Symmetry and Elementary Particles," 1st ed., Academic Press, New York, 1970, Young diagrams are referred to as either 'Young diagrams' or 'Young tableaux'.
337. 154. Young diagrams go by a variety of names in the literature. We follow the nomenclature of G. de B. Robinson, "Representation Theory of the Symmetric Group," University of Toronto Press, Toronto, 1961; A. Messiah, "Quantum Mechanics," Vol. II, North Holland, Amsterdam, 1966; and M. Weissbluth, "Atoms and Molecules," Academic Press, New York, 1978. They also go by the name 'pattern' in Ref. [155]; by the names 'graph', 'diagram', and 'Young Pattern' in Ref. [143]; by the name 'shape' in D. E. Rutherford, "Substitutional Analysis," Edinburgh University Press, Edinburgh, 1948 and C. D. H. Chisholm, op. cit., and by 'frame' in Ref. [149]. Reference [142] reserves the name 'diagram' for Young diagrams and one of the standard Young tableaux. In D. B. Lichtenberg, "Unitary Symmetry and Elementary Particles," 1st ed., Academic Press, New York, 1970, Young diagrams are referred to as either 'Young diagrams' or 'Young tableaux'.
338. 154. Young diagrams go by a variety of names in the literature. We follow the nomenclature of G. de B. Robinson, "Representation Theory of the Symmetric Group," University of Toronto Press, Toronto, 1961; A. Messiah, "Quantum Mechanics," Vol. II, North Holland, Amsterdam, 1966; and M. Weissbluth, "Atoms and Molecules," Academic Press, New York, 1978. They also go by the name 'pattern' in Ref. [155]; by the names 'graph', 'diagram', and 'Young Pattern' in Ref. [143]; by the name 'shape' in D. E. Rutherford, "Substitutional Analysis," Edinburgh University Press, Edinburgh, 1948 and C. D. H. Chisholm, op. cit., and by 'frame' in Ref. [149]. Reference [142] reserves the name 'diagram' for Young diagrams and one of the standard Young tableaux. In D. B. Lichtenberg, "Unitary Symmetry and Elementary Particles," 1st ed., Academic Press, New York, 1970, Young diagrams are referred to as either 'Young diagrams' or 'Young tableaux'.
339. 154. Young diagrams go by a variety of names in the literature. We follow the nomenclature of G. de B. Robinson, "Representation Theory of the Symmetric Group," University of Toronto Press, Toronto, 1961; A. Messiah, "Quantum Mechanics," Vol. II, North Holland, Amsterdam, 1966; and M. Weissbluth, "Atoms and Molecules," Academic Press, New York, 1978. They also go by the name 'pattern' in Ref. [155]; by the names 'graph', 'diagram', and 'Young Pattern' in Ref. [143]; by the name 'shape' in D. E. Rutherford, "Substitutional Analysis," Edinburgh University Press, Edinburgh, 1948 and C. D. H. Chisholm, op. cit., and by 'frame' in Ref. [149]. Reference [142] reserves the name 'diagram' for Young diagrams and one of the standard Young tableaux. In D. B. Lichtenberg, "Unitary Symmetry and Elementary Particles," 1st ed., Academic Press, New York, 1970, Young diagrams are referred to as either 'Young diagrams' or 'Young tableaux'.
340. 155. H. Weyl, "The Theory of Groups and Quantum Mechanics," Dover, New York, 1950.
341. 156. Reference [143, p. 200] or [149, p. 104].
342. 157. See, for example, [143, p. 200] or [149, p. 111].
343. 158. We follow the definition in [143, p. 244; 142, p. 120; 155, p. 359]; M. Weissbluth, op. cit., p. 148; and [150, p. 80]. The definition script capital Y sign = ℘ script capital Q sign is also widely used in the literature (see for example [149] , D. E. Rutherford, op. cit.).
344. 158. We follow the definition in [143, p. 244; 142, p. 120; 155, p. 359]; M. Weissbluth, op. cit., p. 148; and [150, p. 80]. The definition script capital Y sign = ℘ script capital Q sign is also widely used in the literature (see for example [149] , D. E. Rutherford, op. cit.).
345. 158. We follow the definition in [143, p. 244; 142, p. 120; 155, p. 359]; M. Weissbluth, op. cit., p. 148; and [150, p. 80]. The definition script capital Y sign = ℘ script capital Q sign is also widely used in the literature (see for example [149] , D. E. Rutherford, op. cit.).
346. 159. See for example [142, p. 11].
347. 160. D. F. Lawden, "An Introduction to Tensor Calculus and Relativity," 1st ed., Methuen, London, 1962.
348. 161. For a definition of covariant and contravariant tensors see, for example, [142, p. 10; 155, Section 1.3; 160, Section 30].
349. 162. See for example [160, p. 25].
350. 163. G. Arfken, "Mathematical Methods for Physicists," 3rd ed., p. 179, Academic Press, New York, 1985.
351. k is the invariant fundamental tensor. In fact the converse of the above is also true. Requiring that the metric tensor δ the primitive invariant tensor of a group constrains this group to be O(D), see Ref. [152]. (equation presented)
352. 165. The restriction is effected by the addition of new primitive invariant tensors which the subgroup O(D) must respect.
353. 166. References [142, Chap. IV; 149, Chap. V; 143, Chap. 10; 150, Chap. 8].
354. 167. Reference [142, p. 148].
355. 168. Reference [142, Chap. V, Sect. B].
356. 169. Reference [142, Chap. V, Sect. 6].
357. 170. References [143, p. 393; 8, p. 381].
358. n for the one-fold traceless components.
359. 172. Reference [142, Chap. V, Sect. 7].
360. 173. J. Mathews and R. L. Walker, "Mathematical Methods of Physics," Sect. 16.7, Benjamin, Menlo Park, 1970.
361. 174. J. M. Normand and J. Raynal, J. Phys. A 15 (1982), 1437; J. M. Normand, "A Lie Group: Rotations in Quantum Mechanics," North-Holland, Amsterdam, 1980; U. Fano and G. Racah, "Irreducible Tesorial Sets," Academic Press, New York, 1959.
362. 174. J. M. Normand and J. Raynal, J. Phys. A 15 (1982), 1437; J. M. Normand, "A Lie Group: Rotations in Quantum Mechanics," North-Holland, Amsterdam, 1980; U. Fano and G. Racah, "Irreducible Tesorial Sets," Academic Press, New York, 1959.
363. 174. J. M. Normand and J. Raynal, J. Phys. A 15 (1982), 1437; J. M. Normand, "A Lie Group: Rotations in Quantum Mechanics," North-Holland, Amsterdam, 1980; U. Fano and G. Racah, "Irreducible Tesorial Sets," Academic Press, New York, 1959.
364. n | LMα)†; see the paragraph of Appendix A in which Eq. (A8) is to be found, i.e., the Clebsch-Gordon coefficients will not in general effect a unitary change of basis.
365. t commute. (equation presented)
366. 177. See, for example, [150, Sects. 6.2 and 6.3].
367. 178. A more direct approach for using the primitive invariant tensors to reduce tensor spaces, along with a useful graphical calculus to assist in this endeavor, has been sponsored by Cvitanović and others [152] and has the advantage of decomposing tensor spaces into orthogonal irreducible sub-spaces. This approach, though, is not as well suited to deriving general results for tensors of arbitrary rank as the approach outlined in Section II.
368. n are generally not symmetric. (equation presented)
369. 180. Reference [142, Theorem (5.7.A), p. 154].
370. 181. Reference [142, Theorems (5.7.D) and (5.7.E), p. 157].
371. 181. Reference [142, Theorems (5.7.D) and (5.7.E), p. 157].
372. 182. See Reference [142, p. 150, 152, and 158]; M. Dunn and D. K. Watson, unpublished.
373. 182. See Reference [142, p. 150, 152, and 158]; M. Dunn and D. K. Watson, unpublished.
374. D-1) [186].
375. 184. R. Courant and D. Hubert, "Methods of Mathematical Physics," Vol. 1, p. 65, Interscience, New York, 1953.
376. 185. R. D. Richtmyer, "Principles of Advanced Mathematical Physics," Vol. 1, Chap. 5, Springer-Verlag, New York, 1978.
377. 186. Ibid, p. 79.
378. 187. See, for example, A. B. Wolbarst, "Symmetry and Quantum Systems," p. 68, Van Nostrand Reinhold, London, 1977.
379. 188. Z. Y. Wen and J. Avery, J. Math. Phys. 26 (1985), 396. There are more general formulas for the number of independent components of an irreducible tensor of O(D), however, they are not needed to achieve this identification. See D. E. Littlewood, "The Theory of Group Characters," Eq. (11.8; 6), p. 236, Oxford University Press, Oxford, 1950; R. C. King, Can. J. Math. 33 (1972), 176; G. Murtaza and M. A. Rashid, J. Math. Phys. 14 (1973), 1196.
380. 188. Z. Y. Wen and J. Avery, J. Math. Phys. 26 (1985), 396. There are more general formulas for the number of independent components of an irreducible tensor of O(D), however, they are not needed to achieve this identification. See D. E. Littlewood, "The Theory of Group Characters," Eq. (11.8; 6), p. 236, Oxford University Press, Oxford, 1950; R. C. King, Can. J. Math. 33 (1972), 176; G. Murtaza and M. A. Rashid, J. Math. Phys. 14 (1973), 1196.
381. 188. Z. Y. Wen and J. Avery, J. Math. Phys. 26 (1985), 396. There are more general formulas for the number of independent components of an irreducible tensor of O(D), however, they are not needed to achieve this identification. See D. E. Littlewood, "The Theory of Group Characters," Eq. (11.8; 6), p. 236, Oxford University Press, Oxford, 1950; R. C. King, Can. J. Math. 33 (1972), 176; G. Murtaza and M. A. Rashid, J. Math. Phys. 14 (1973), 1196.
382. 188. Z. Y. Wen and J. Avery, J. Math. Phys. 26 (1985), 396. There are more general formulas for the number of independent components of an irreducible tensor of O(D), however, they are not needed to achieve this identification. See D. E. Littlewood, "The Theory of Group Characters," Eq. (11.8; 6), p. 236, Oxford University Press, Oxford, 1950; R. C. King, Can. J. Math. 33 (1972), 176; G. Murtaza and M. A. Rashid, J. Math. Phys. 14 (1973), 1196.
383. 189. This has important consequences for the MO description of doubly excited states of two electron atoms [84, 85] and the Born-Oppenheimer description of molecules in D dimensions, Λ-doubling (l-doubling) is broken when D ≠ 3, even in these approximations, although the magnitude of the splitting is small when D is small [85].
384. tn, which may or may not be complete and would not lead to the expansion that we wish to achieve. (equation presented)
385. 191. R. R. Stoll, "Linear Algebra and Matrix Theory," p. 151, McGraw-Hill, New York, 1952.
386. 192. Ibid., p. 154.
387. 193. Sylvesters Law: see, for example, P. Roman, "Some Modern Mathematics for Physicists and Other Outsiders," Vol. 1, p. 139, Pergamon, New York, 1975.
388. 194. See for example [143, p. 100] or [149, p. 22].
389. 195. This last point is a consequence of Sylvesters Law; see P. Roman, op. cit., Theorem 2.1(1), p. 35; Theorem 4.3b(2), p. 139; and Theorem 4.3b(3), p. 140.
390. 195. This last point is a consequence of Sylvesters Law; see P. Roman, op. cit., Theorem 2.1(1), p. 35; Theorem 4.3b(2), p. 139; and Theorem 4.3b(3), p. 140.
391. 195. This last point is a consequence of Sylvesters Law; see P. Roman, op. cit., Theorem 2.1(1), p. 35; Theorem 4.3b(2), p. 139; and Theorem 4.3b(3), p. 140.
392. 196. Reference [149, p. 7].
393. 197. See, for example, [142, p. 102].
394. 198. See, for example, [149, p. 53].
395. 199. See, for example, [149, p. 55].
396. 200. See, for example, [143, p. 107] or [199].
397. 201. See, for example, [143, p. 107] or [149, p. 70].
398. 202. See, for example, [143, p. 243] or [149, p. 59].
399. 203. Reference [142, p. 104].
400. 204. See, for example, [149, p. 61].
401. 205. See, for example, [149, p. 62].
402. 206. See, for example, [149, p. 63].
403. 207. See, for example, [142, p. 103].
404. 208. See, for example, [149, p. 66].
405. 209. See, for example, [149, p. 64].
406. 210. This relationship is formalized by Wedderburn's theorem [142, p. 90; 149, p. 67] and [142, Theorem (3.5), p. 93] or [149, Theorem 8.2, p. 23].
407. 211. Reference [149, p. 114].
408. 212. See, for example, [149, p. 105].
409. 213. D. E. Rutherford, op. cit., p. 21.
410. 214. Although [149, p. 113] considers Young operators of the form script capital Y sign = ℘ script capital Q sign, only minor modifications of this argument are required to prove the result quoted in this paper.
411. 215. See, for example, [149, p. 113].
412. 216. See, for example, [142, p. 124].
413. 217. See, for example, [142, p. 125].
414. 218. See, for example, [142, p. 121].
415. 219. See, for example, [143, p. 110].