Theoretical and Mathematical Physics

, Volume 87, Issue 3, pp 561–599 | Cite as

Splitting of the lowest energy levels of the Schrödinger equation and asymptotic behavior of the fundamental solution of the equation hut=h2Δu/2−V(x)u

  • S. Yu. Dobrokhotov
  • V. N. Kolokol'tsov
  • V. P. Maslov


Lower Energy Energy Level Asymptotic Behavior Fundamental Solution Lower Energy Level 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literature Cited

  1. 1.
    V. P. Maslov, Dokl. Akad. Nauk SSSR,258, 1112 (1981).Google Scholar
  2. 2.
    V. P. Maslov, Tr. Mosk. Inst. Akad. Nauk SSSR,163, 150 (1984).Google Scholar
  3. 3.
    V. P. Maslov, Perturbation Theory and Asymptotic Methods [in Russian], Moscow State University, Moscow (1965).Google Scholar
  4. 4.
    V. P. Maslov, Asymptotic Methods and Perturbation Theory [in Russian], Nauka, Moscow (1988).Google Scholar
  5. 5.
    V. P. Maslov, The Complex WKB Method in Nonlinear Equations [in Russian], Nauka, Moscow (1977).Google Scholar
  6. 6.
    V. P. Maslov, Zh. Vychisl. Mat. Mat. Fiz., No. 1, 114 (1961); No. 4, 638 (1961).Google Scholar
  7. 7.
    S. R. Varadhan, Commun. Pure Appl. Math.,19, 231 (1966).Google Scholar
  8. 8.
    A. A. Borovkov, Teor. Veroyatn. Ee Primen.,12, 635 (1967).Google Scholar
  9. 9.
    B. Simon, Ann. Inst. H. Poincaré,38, 295 (1983).Google Scholar
  10. 10.
    A. M. Polyakov, Nucl. Phys.,120, 429 (1977).Google Scholar
  11. 11.
    E. Gildener and A. Patrascigin, Phys. Rev. D,16, 425 (1977).Google Scholar
  12. 12.
    E. M. Harrel, Commun. Math. Phys.,75, 239 (1980).Google Scholar
  13. 13.
    E. M. Harrel, Ann. Phys. (N. Y.),119, 351 (1979).Google Scholar
  14. 14.
    G. Jona-Lasinio, L. Martinelli, and E. Scoppola, Commun. Math. Phys.,80, (1981).Google Scholar
  15. 15.
    J. M. Combes, P. Duclos, and R. Seiler, Commun. Math. Phys.,92, No. 2 (1983).Google Scholar
  16. 16.
    T. F. Pankratova, Dokl. Akad. Nauk SSSR,276, 795 (1984).Google Scholar
  17. 17.
    R. Rajaraman, Solitons and Instantons, North-Holland, Amsterdam (1984).Google Scholar
  18. 18.
    B. Simon, Ann. Math.,120, 89 (1984).Google Scholar
  19. 19.
    B. Heffler and J. Sjöstrand, Commun. in P. D. E.,9, 337 (1984); Ann. Inst. H. Poincaré,42, 127 (1985); Commun. in P. D. E.,13, 245 (1985); Ann. Inst. H. Poincaré,46, 353 (1987).Google Scholar
  20. 20.
    E. Witten, J. Diff. Geom.,17, 661 (1982).Google Scholar
  21. 21.
    L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Nonrelativistic Theory, 3rd ed., Pergamon Press, Oxford (1977).Google Scholar
  22. 22.
    P. Hartman, Ordinary Differential Equations, Wiley, New York (1964).Google Scholar
  23. 23.
    Yu. I. Kifer, Izv. Akad. Nauk SSSR, Ser. Mat.,38, 1091 (1974).Google Scholar
  24. 24.
    V. M. Babich, Dokl. Akad. Nauk SSSR,289, 836 (1986).Google Scholar
  25. 25.
    S. A. Molchanov, Usp. Mat. Nauk,30, 3 (1975).Google Scholar
  26. 26.
    V. P. Maslov, Complex Markov Chains and Feynman Path Integrals [in Russian], Nauka, Moscow (1977).Google Scholar
  27. 27.
    V. P. Maslov and M. V. Fedorov, The Semiclassical Approximation for the Equations of Quantum Mechanics [in Russian], Nauka, Moscow (1976).Google Scholar
  28. 28.
    V. N. Kolokol'tsov and V. P. Maslov, Funktsional. Analiz i Ego Prilozhen., No. 1, 1 (1989); No. 4, 53 (1989).Google Scholar
  29. 29.
    V. P. Maslov, Asymptotic Methods of Solution of Pseudodifferential Equations [in Russian], Nauka, Moscow (1987).Google Scholar
  30. 30.
    V. P. Maslov, Usp. Mat. Nauk,42, 39 (1987).Google Scholar
  31. 31.
    J. Palis, Jr, and W. de Melo, Geometric Theory of Dynamical Systems, Springer, New York (1982).Google Scholar
  32. 32.
    S. Sternberg, Am. J. Math.,79, 809 (1957).Google Scholar
  33. 33.
    F. A. Berezin and M. A. Shubin, The Schrödinger Equation [in Russian], Moscow State University, Moscow (1983).Google Scholar
  34. 34.
    V. P. Maslov and M. V. Fedoryuk, Mat. Zametki.,30, 763 (1981).Google Scholar
  35. 35.
    E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, Vol. 2, Clarendon Press, Oxford (1958).Google Scholar
  36. 36.
    M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 4, Academic Press, New York (1978).Google Scholar
  37. 37.
    S. Coleman, “The uses of instantons,” in: The Whys of Subnuclear Physics (ed. A. Zichichi), New York (1979), pp. 805–916.Google Scholar
  38. 38.
    O. Madelung, Theory of Solids [Russian translation], Nauka, Moscow (1980).Google Scholar
  39. 39.
    S. S. Gershtein, L. I. Ponomarev, and T. P. Puzynina, Zh. Eksp. Teor. Fiz.,48, No. 2 (1965).Google Scholar
  40. 40.
    M. V. Fedoryuk, Asymptotics, Integrals, and Series [in Russian], Nauka, Moscow (1987).Google Scholar

Copyright information

© Plenum Publishing Corporation 1991

Authors and Affiliations

  • S. Yu. Dobrokhotov
  • V. N. Kolokol'tsov
  • V. P. Maslov

There are no affiliations available

Personalised recommendations