Skip to main content
SpringerLink
Log in

Semiclassical Trajectory-Coherent Approximations of Hartree-Type Equations

  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

Abstract

We use the concept of the complex WKB–Maslov method to construct semiclassically concentrated solutions for Hartree-type equations. Formal solutions of the Cauchy problem for this equation that are asymptotic (with respect to a small parameter ℏ, ℏ→0) are constructed with the power-law accuracy O(ℏN/2), where N≥3 is a positive integer. The system of Hamilton–Ehrenfest equations (for averaged and centered moments) derived in this paper plays a significant role in constructing semiclassically concentrated solutions. In the class of semiclassically concentrated solutions of Hartree-type equations, we construct an approximate Green's function and state a nonlinear superposition principle.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. M. V. Karasev and V. P. Maslov, Nonlinear Poisson Brackets: Geometry and Quantization [in Russian], Nauka, Moscow (1991); English transl., Am. Math. Soc., Providence, R. I. (1993).

    Google Scholar 

  2. S. P. de Groot and L. G. Suttirp, Foundations of Electrodynamics, North–Holland, Amsterdam (1972).

    Google Scholar 

  3. S. I. Pekar, Studies in the Electronic Theory of Crystals [in Russian], Gostekhizdat, Moscow (1951).

    Google Scholar 

  4. P. Ring and P. Schuck, The Nuclear Many-Body Problem, New York (1980).

  5. J. Kurlandski, Bull. Acad. Pol. Sci., 30, No. 3–4, 135 (1982).

    Google Scholar 

  6. V. M. Ol'khov, Theor. Math. Phys., 51, 414 (1982).

    Google Scholar 

  7. D. V. Chudnovsky, “Infinite component two-dimensional completely integrable systems of KdV type,” in: The Riemann Problem, Complete Integrability, and Arithmetic Applications (Lect. Notes Math., Vol. 925, D. Chudnovsky and G. Chudnovsky, eds.), Springer, Berlin (1982), p. 71.

    Google Scholar 

  8. A. A. Bogolyubskaya and I. L. Bogolyubskii, Theor. Math. Phys., 54, 168 (1983).

    Google Scholar 

  9. D. Gogny and P. L. Lions, J. Math. Phys., 27, 211 (1986).

    Google Scholar 

  10. D. R. Hartree, Proc. Cambridge Philos. Soc., 24, 89, 111, 426 (1928).

    Google Scholar 

  11. G. Efimov, Nonlocal Interaction of Quantum Fields [in Russian], Nauka, Moscow (1977).

    Google Scholar 

  12. F. A. Berezin, Methods of Second Quantization [in Russian], Nauka, Moscow (1965); English transl., Acad. Press, New York (1966, 1986).

    Google Scholar 

  13. A. S. Davydov, Solitons in Molecular Systems [in Russian], Naukova Dumka, Kiev (1984); English transl., Reidel, Dordrecht (1985).

    Google Scholar 

  14. P. N. Brusov and V. N. Popov, Superfluidity and Collective Properties of Quantum Liquids [in Russian], Nauka, Moscow (1988).

    Google Scholar 

  15. Y. Lai and H. A. Haus, Phys. Rev. A, 40, 844, 854 (1989).

    Google Scholar 

  16. D. E. Fillips and M. D. Lukin, Phys. Rev. Lett., 86, 783 (2001).

    Google Scholar 

  17. A. Zozulya, S. Diddams, and T. Clement, Phys. Rev. A, 58, No. 4, 72 (1988).

    Google Scholar 

  18. A. Bove, G. Da Prato, and G. Fano, Commun. Math. Phys., 37, 183 (1974).

    Google Scholar 

  19. J. M. Chadam and R. T. Glassey, J. Math. Phys., 16, 1122 (1975).

    Google Scholar 

  20. R. T. Glassey, Commun. Math. Phys., 53, 9 (1977).

    Google Scholar 

  21. V. Delgado, Proc. Am. Math. Soc., 69, 289 (1978).

    Google Scholar 

  22. I. Fukuda and M. Tsutsumi, J. Math. Anal. Appl., 66, 358 (1978).

    Google Scholar 

  23. E. B. Davies, Ann. Inst. H. Poincaré A, 31, 319 (1979).

    Google Scholar 

  24. J. Ginibre and G. Velo, Math. Z., 170, No. 2, 109 (1980).

    Google Scholar 

  25. Y. Choquet-Bruhat, C. R. Acad. Sci. Ser. 1, 292, No. 2, 153 (1981).

    Google Scholar 

  26. W. A. Strauss, J. Funct. Anal., 43, 281 (1981).

    Google Scholar 

  27. G. P. Menzala and W. A. Strauss, Diff Equat., 43, 93 (1982).

    Google Scholar 

  28. K. Nakamitsu and M. Tsutsumi, J. Math. Phys., 27, 211 (1986).

    Google Scholar 

  29. M. Reeken, J. Math. Phys., 11, 2505 (1970).

    Google Scholar 

  30. K. Gustafson and D. Sather, Rand. Math., 4, 723 (1971).

    Google Scholar 

  31. J. Wolkowsky, Indiana Univ. Math. J., 22, 551 (1972).

    Google Scholar 

  32. C. Stuart, Arch. Rat. Mech. Anal., 51, 60 (1973).

    Google Scholar 

  33. G. Fonte, R. Mignani, and G. Schiffrer, Commun. Math. Phys., 33, 293 (1973).

    Google Scholar 

  34. E. H. Lieb and B. Simon, Commun. Math. Phys., 53, No. 3, 185 (1977).

    Google Scholar 

  35. E. H. Lieb, Stud. Appl. Math., 57, 93 (1977).

    Google Scholar 

  36. P. Bader, Proc. Roy. Soc. Edinburgh. A, 82, 27 (1978).

    Google Scholar 

  37. G. P. Menzala, “On a Hartree-type equation: existence of regular solutions,” in: Functional Differential Equations and Bifurcations (Lect. Notes Math., Vol. 799, A. F. Izé, ed.), Springer, Berlin (1980), p. 277.

    Google Scholar 

  38. G. Rosensteel and E. Ihrig, J. Math. Phys., 21, 2297 (1980).

    Google Scholar 

  39. P. L. Lions, Nonlinear Anal. Theor. Meth. Appl., 4, 1063 (1980).

    Google Scholar 

  40. A. Bongers, Z. Angew. Math. Mech., 60, No. 7, 240 (1980).

    Google Scholar 

  41. J. D. Gegenberg and A. J. Das, J. Math. Phys., 22, 1736 (1981).

    Google Scholar 

  42. P. L. Lions, Nonlinear Anal. Theor. Meth. Appl., 5, 1245 (1981).

    Google Scholar 

  43. H. J. Efinger and H. Grosse, Lett. Math. Phys., 8, 91 (1984).

    Google Scholar 

  44. P. L. Lions, Commun. Math. Phys., 109, 33 (1987).

    Google Scholar 

  45. V. P. Maslov, Complex Markov Chains and the Feynman Path Integral [in Russian], Nauka, Moscow (1976).

    Google Scholar 

  46. V. P. Maslov, “Equations of a self-consistent.eld [in Russian],” in: Contemporary Problems in Mathematics (R. V. Gamkrelidze, ed.), Vol. 11, VINITI, Moscow (1978), p. 153.

    Google Scholar 

  47. M. V. Karasev and V. P. Maslov, “Algebras with general commutation relations and their applications [in Russian],” in: Contemporary Problems in Mathematics (R. V. Gamkrelidze, ed.), Vol. 13, VINITI, Moscow (1979), p. 145.

    Google Scholar 

  48. V. P. Maslov, The Complex WKB Method for Nonlinear Equations [in Russian], Nauka, Moscow (1977).

    Google Scholar 

  49. I. V. Simenog, Theor. Math. Phys., 30, 263 (1977).

    Google Scholar 

  50. S. A. Vakulenko, V. P. Maslov, I. A. Molotkov, and A. I. Shafarevich, Dokl. Math., 52, 456 (1995).

    Google Scholar 

  51. I. A. Molotkov, S. A. Vakulenko, and M. A. Bisyarin, Nonlinear Localized Wave Processes [in Russian], Yanus-K, Moscow (1999).

    Google Scholar 

  52. S. I. Chernykh, Theor. Math. Phys., 52, 939 (1982).

    Google Scholar 

  53. M. V. Karasev and A.V. Pereskokov, Theor. Math. Phys., 79, 479 (1989).

    Google Scholar 

  54. M. V. Karasev and A.V. Pereskokov, Theor. Math. Phys., 97, 1160 (1993).

    Google Scholar 

  55. M. V. Karasev and A.V. Pereskokov, Izv. Rossiiskoi Akad. Nauk, Ser. Mat., 65, No. 5, 33 (2001).

    Google Scholar 

  56. M. V. Karasev and A.V. Pereskokov, Izv. Rossiiskoi Akad. Nauk, Ser. Mat., 65, No. 6, 57 (2001).

    Google Scholar 

  57. V. P. Maslov, Russ. J. Math. Phys., 3, No. 2, 1 (1995).

    Google Scholar 

  58. V. P. Maslov and O. Yu. Shvedov, The Complex Germ Method in Quantum Field Theory [in Russian], Izd. URSS, Moscow (1998).

    Google Scholar 

  59. V. G. Bagrov, V. V. Belov, and I. M. Ternov, Theor. Math. Phys., 50, 256 (1982).

    Google Scholar 

  60. V. G. Bagrov, V. V. Belov, and I. M. Ternov, J. Math. Phys., 24, 2855 (1983).

    Google Scholar 

  61. V. G. Bagrov, V. V. Belov, and A. Yu. Trifonov, “Semiclassically concentrated states of the Schrödinger equation [in Russian],” in: Lecture Notes in Theoretical and Mathematical Physics, Vol. 1, Part 1 (A. V. Aminova, ed.), Kazan State Univ., Kazan (1996), p. 15.

    Google Scholar 

  62. V. G. Bagrov, V. V. Belov, and A. Yu. Trifonov, Ann. Phys., 246, 231 (1996).

    Google Scholar 

  63. V. V. Belov and V. P. Maslov, Sov. Phys. Dokl., 34, 220 (1989).

    Google Scholar 

  64. V.V. Belov and V. P. Maslov, Sov. Phys. Dokl., 35, 330 (1990).

    Google Scholar 

  65. V.V. Belov and M. F. Kondrat'eva, Theor. Math. Phys., 92, 722 (1992).

    Google Scholar 

  66. E. Schrödinger, Naturwissenschaften, 14, 664 (1926).

    Google Scholar 

  67. R. J. Glauber, Phys. Rev., 130, 2529 (1963); 131, 2766 (1963).

    Google Scholar 

  68. P. K. Rashevskii, Usp. Mat. Nauk, 13, No. 3, 3 (1958).

    Google Scholar 

  69. J. R. Klauder, J. Math. Phys., 4, 1055, 1058 (1963); 5, 177 (1964).

    Google Scholar 

  70. N. A. Chernikov, JETP, 53, 1006 (1967).

    Google Scholar 

  71. M. A. Malkin and V. I. Man'ko, Dynamic Symmetries and Coherent States of Quantum Systems [in Russian], Nauka, Moscow (1979).

    Google Scholar 

  72. A. M. Perelomov, Generalized Coherent States and Their Applications, Nauka, Moscow (1987); English transl., Springer, Berlin (1986).

    Google Scholar 

  73. V. V. Belov, “Semiclassical limit for equations of motion of quantum averages in a nonrelativistic system with gauge fields [in Russian],” Preprint No. 58, Tomsk Sci. Center, Siberian Branch, Acad. Sci. USSR, Tomsk (1989).

    Google Scholar 

  74. V. G. Bagrov, V. V. Belov, M. F. Kondratyeva, A. M. Rogova, and A. Yu. Trifonov, J. Moscow Phys. Soc., 3, 309 (1993).

    Google Scholar 

  75. V. G. Bagrov, V. V. Belov, and M. F. Kondrat'eva, Theor. Math. Phys., 98, 34 (1994).

    Google Scholar 

  76. V. V. Belov and M. F. Kondrat'eva, Math. Notes, 56, 1228 (1995).

    Google Scholar 

  77. V. V. Belov and M. F. Kondrat'eva, Math. Notes, 58, 1251 (1995).

    Google Scholar 

  78. V. P. Maslov, The Complex WKB Method for Nonlinear Equations: I. Linear Theory, Birkhäuser, Basel (1994).

    Google Scholar 

  79. H. Hayashi and P. I. Naumkin, SUT J. Math., 34, No. 1, 13 (1998).

    Google Scholar 

  80. J. Ginibre and G. Velo, Rev. Math. Phys., 12, 361 (2000); Ann. Inst. H. Poincaré, vn1, pp753 (2000).

    Google Scholar 

  81. V. P. Maslov, Operator Methods [in Russian], Nauka, Moscow (1973); English transl.: Operational Methods, Mir, Moscow (1976).

    Google Scholar 

  82. H. Bateman and A. Erdélyi, eds., Higher Transcendental Functions (Based on notes left by H. Bateman), Vol. 2, McGraw-Hill, New York (1953).

    Google Scholar 

  83. V. I. Arnold, Mathematical Methods of Classical Mechanics [in Russian] (2nd ed.), Nauka, Moscow (1983); English transl. prev. ed., Springer, Berlin (1978).

    Google Scholar 

  84. V. M. Babich and V. S. Buldyrev, Asymptotic Methods in Short–Wavelength Diffraction Theory [in Russian], Nauka, Moscow (1972); English transl.: Short–Wavelength Diffraction Theory: Asymptotic Methods, Springer, Berlin (1991).

    Google Scholar 

  85. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 3, Scattering Theory, Acad. Press, New York (1979).

    Google Scholar 

  86. M. M. Popov, “Green's functions of the Schrödinger equation with a quadratic potential [in Russian],” in: Problems in Mathematical Physics (V. M. Babich, ed.), Izd. Leningrad. Univ., Leningrad (1973), p. 119.

    Google Scholar 

  87. V. V. Dodonov, I. A. Malkin, and V. I. Man'ko, Int. J. Theor. Phys., 14, 37 (1975).

    Google Scholar 

  88. V. V. Belov, G. N. Serezhnikov, A. Yu. Trifonov, and A. V. Shapovalov, “Symmetry and differential equations [in Russian],” in: Proc. Intern. Conf. Krasnoyarsk, 21–25 August2000, Krasnoyarsk State Univ., Krasnoyarsk (2000), p. 39.

    Google Scholar 

  89. V. D. Baranov, V. V. Belov, A. Yu. Trifonov, and A. V. Shapovalov, “Latest problem in field theory [in Russian],” in: Proc. Intl. Summer School–Seminar in Modern Problems in Theor. and Math. Physics, Kazan, 2000 (A. V. Aminova, ed.), p. 22.

  90. G. N. Serezhnikov, A. Yu. Trifonov, and A. V. Shapovalov, Izv. Tomsk. Gos. Univ., 1, 39 (2000).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Moscow Institute for Electronics and Mathematics, Moscow, Russia

    V. V. Belov

  2. Tomsk Polytechnical University, Tomsk, Russia

    A. Yu. Trifanov

  3. Tomsk State University, Tomsk, Russia

    A. V. Shapovalov

Authors
  1. V. V. Belov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Yu. Trifanov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. V. Shapovalov
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Belov, V.V., Trifanov, A.Y. & Shapovalov, A.V. Semiclassical Trajectory-Coherent Approximations of Hartree-Type Equations. Theoretical and Mathematical Physics 130, 391–418 (2002). https://doi.org/10.1023/A:1014719007121

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1014719007121

Keywords

Search

Navigation