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Prog. Theor. Phys. Supplement No.131 (1998) pp. 459-470

[ Full Text PDF : FREE ACCESS (564K) ]

Renormalization-Group Resummation of a Divergent Series of the Perturbative Wave Functions of Quantum Systems

Teiji Kunihiro*

Faculty of Science and Technology, Ryukoku University, Otsu 520-2194, Japan

Abstract:

The perturbative renormalization group (RG) equation is applied to resum a divergent series of perturbative wave functions of a quantum anharmonic oscillator. It is found that the resummed series gives the cumulant of the naive perturbation series. It is shown that a reorganization of the resummed series reproduce the correct asymptotic form of the wave function at x→∞ when the perturbation expansion is stopped at the fourth order. A brief comment is given on the relation between the present method and the delta-expansion method, which is based on a kind of nonperturbative RG equation.


URL : http://ptp.ipap.jp/link?PTPS/131/459/
DOI : 10.1143/PTPS.131.459


*E-mail address: kunihiro@rins.ryukoku.ac.jp

[ Full Text PDF : FREE ACCESS (564K) ] Citation:

References:

  1. E. C. G. Stuckelberg and A. Petermann, Helv. Phys. Acta 26 (1953), 499.
    M. Gell-Mann and F. E. Low, Phys. Rev. 95 (1953), 1300[APS].
    See also K. Wilson, Phys. Rev. D3 (1971), 1818[APS],
    S. Weinberg, in Asymptotic Realms of Physics, ed. A. H. Guth et al. (MIT Press, 1983); The Quantum Theory of Fields (Cambridge University Press, 1996), vol. II, chap. 18.
  2. N. Goldenfeld, O. Martin and Y. Oono, J. Sci. Comp. 4 (1989), 4.
    N. Goldenfeld , O. Martin, Y. Oono and F. Liu, Phys. Rev. Lett. 64 (1990), 1361[APS].
    N. D. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group (Addison-Wesley, Reading, Mass., 1992).
    L. Y. Chen, N. Goldenfeld, Y. Oono and G. Paquette, Physica A204 (1994), 111.
    G. Paquette, L. Y. Chen, N. Goldenfeld and Y. Oono, Phys. Rev. Lett. 72 (1994), 76[APS].
    L.-Y. Chen, N. Goldenfeld and Y. Oono, Phys. Rev. Lett. 73 (1994), 1311[APS]; Phys. Rev. E54 (1996), 376[APS].
  3. C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978).
  4. T. Kunihiro, Prog. Theor. Phys. 94 (1995), 503 [PTP][Errata; 95 (1996), 835[PTP]]; Jpn. J. Ind. Appl. Math. 14 (1997), 51.
  5. T. Kunihiro, Prog. Theor. Phys. 97 (1997), 179, [PTP]
    see also, patt-sol/9709003, on which §2.1 of the present report is based.
  6. S. Coleman and E. Weinberg, Phys. Rev. D7 (1973), 1888[APS].
    M. Sher, Phys. Rep. 179 (1989), 274.
    M. Bando, T. Kugo, N. Maekawa and H. Nakano, Phys. Lett. B301 (1993), 83[Elsevier]; Prog. Theor. Phys. 90 (1993), 405[PTP].
    C. Ford, Phys. Rev. D50 (1994), 7531[APS].
    H. Nakkagawa and H. Yokota, Modern Phys. Lett. A11 (1996), 2259.
  7. D. V. Shirkov, Russian Math. Surveys 49:5 (1994), 155; hep-th/9602024[e-print arXiv].
    V. F. Kovalev, V. V. Pustovalov and D. V. Shirkov, hep-th/9706056[e-print arXiv], to be published in J. Math. Phys.
  8. Large Order Behavior of Perturbation Theory, Current Physics – Sources and Comments, vol. 7, ed. J. C. Le Guillou and J. Zinn-Justin (North-Holland, Amsterdam, 1990).
    G. A. Arteca, F. M. Fernández and E. A. Castro, Large Order Perturbation Theory and Summation Methods in Quantum Mechanics (Springer-Verlag, Berlin, 1990).
  9. T. Kunihiro, Phys. Rev. D57 (1998), R2035[APS].
  10. C. M. Bender and L. M. A. Bettencourt, Phys. Rev. Lett. 77 (1996), 4114[APS]; Phys. Rev. D54 (1996), 7710[APS].
  11. M. Frasca, Nuovo Cim. B107 (1992), 915; ibid. B109 (1994), 603.
  12. C. A. Ginsburg and E. W. Montroll, J. Math. Phys. 19 (1978), 336[AIP Scitation].
  13. I. L. Egusquiza and M. A. Valle Basagoiti, hep-th/9611143[e-print arXiv].
  14. M. Frasca, Phys. Rev. A56 (1997), 1548[APS].
  15. R. Graham, Phys. Rev. Lett. 76 (1996), 2185[APS].
    S. Sasa, Physica D108 (1997), 45.
    H. J. de Vega and J. F. J Salgado, Phys. Rev. D56 (1997), 6524[APS].
    T. Kunihiro and J. Matsukidaira, Phys. Rev. E57 (1998), 4817[APS].
  16. For example, see R. Kubo, M. Toda and N. Hashitsume, Statistical Physics II Nonequilibrium Statistical Mechanics, 2nd ed. (Springer 1995).
  17. C. M. Bender and T. T. Wu, Phys. Rev. 184 (1969), 1231[APS]; ibid. D7 (1973), 1620[APS].
  18. P. M. Stevenson, Phys. Rev. D23 (1981), 2916[APS].
    C. Arvanitis, H. F. Jones and C. S. Parker, Phys. Rev. D52 (1995), 3704[APS].
    H. Kleinert, Path Integrals in Quantum Mechanics, Statistical and Polymer Physics, 2nd. ed. (World Scientific, Singapore, 1995), and references cited therein.
    E. J. Weniger, Phys. Rev. Lett. 77 (1996), 2859[APS].
    R. Guida, K. Konishi and H. Suzuki, Ann. of Phys. 241 (1995), 152[CrossRef]; ibid. 249 (1996), 109[CrossRef].
  19. T. Hatsuda, T. Kunihiro and T. Tanaka, Phys. Rev. Lett. 78 (1997), 3229[APS].
    T. Tanaka, Phys. Lett. A (1998), to be published.
    S. K. Kauffman and S. M. Perez, J. of Phys. A17 (1984), 2027[IoP STACKS]; ibid. A19 (1986), 3807[IoP STACKS].

Citing Article(s) :

  1. Progress of Theoretical Physics Vol. 108 No. 3 (2002) pp. 571-590 :
    Non-Perturbative Renormalization Group Analysis in Quantum Mechanics
    Ken-Ichi Aoki, Atsushi Horikoshi, Masaki Taniguchi and Haruhiko Terao
  2. Progress of Theoretical Physics Vol. 126 No. 5 (2011) pp. 761-809 :
    First-Principles Derivation of Stable First-Order Generic-Frame Relativistic Dissipative Hydrodynamic Equations from Kinetic Theory by Renormalization-Group Method
    Kyosuke Tsumura and Teiji Kunihiro

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