Advertisement

Communications in Mathematical Physics

, Volume 73, Issue 3, pp 247–264 | Cite as

Sur une Resolution Stochastique de l'Equation de Schrödinger à Coefficients Analytiques

  • Halim Doss
Article

Abstract

Under some hypotheses of analyticity and integrability we show the existence and uniqueness of a strong regular solution of the Schrödinger equation using a natural generalisation to the complex case of the Feynman-Kac formula. This explicit representation allows us to study in certain cases the asymptotic behavior of the solution when the Planck constanth tends to zero. The same method can be used for the solution of more general Schrödinger equations.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliographie

  1. 1.
    Albeverio, S.A., Høegh-Krohn, R.J.: Mathematical theory of Feynman path integrals. Lecture notes in mathematics523. Berlin, Heidelberg, New York: Springer 1976Google Scholar
  2. 2.
    Albeverio, S.A., Høegh-Krohn, R.J.: Oscillatory integrals and the method of stationary phase in infinitely many dimensions, with applications to the classical limit of quantum mechanics. I. Inventiones Maths.106, 40–49 (1977)Google Scholar
  3. 3.
    de Broglie, L.: Recherches d'un demi-siècle. Paris: Editions Albin Michel 1976Google Scholar
  4. 4.
    Caméron, R.H.: A family of integrals serving to connect the Wiener and Feynman integrals. J. Math. Phys.39, 126–141 (1961)Google Scholar
  5. 5.
    Caméron, R.H.: The Ilstow and Feynman integrals. J. d'Anal. Math.10, 187–361 (1962–63)Google Scholar
  6. 6.
    Doss, H.: Quelques formules asymptotiques pour les petites perturbations de systèmes dynamiques. Ann. Inst. Henri Poincaré (à paraître)Google Scholar
  7. 7.
    Doss, H.: Liens entre équations différentielles stochastiques et ordinaires. Ann. Inst. Henri PoincaréXIII, Section B, 99–125 (1977)Google Scholar
  8. 8.
    Feynman, R.P., Hibbs, A.R.: Quantum mechanics and path integrals. New York: MacGraw Hill 1968Google Scholar
  9. 9.
    Gelfand, I.M., Yaglom, A.M.: Integration in functional spaces and it's applications in quantum physics. J. Math. Phys.1, 48–69 (1960)Google Scholar
  10. 10.
    Itô, K.: Wiener integrals and Feynman integral. Proc. Fourth Berkeley Symp. on Math. and Prob. Vol. 2, pp. 227–238. Berkeley: Univ. California Press 1961Google Scholar
  11. 11.
    Itô, K.: Generalized uniform complex measures in the Hilbertian metric space with their application to the Feynman path integral. Proc. Fifth Berkeley Symp. on Math. Stat. and Prob. Vol. 11, pp. 145–161. Berkeley: Univ. California Press 1967Google Scholar
  12. 12.
    Kac, M.: Probability and related topics in physical sciences. New York: Interscience 1959Google Scholar
  13. 13.
    Kunita, H.: Diffusion processes and control systems. Cours de 3ème Cycle, Second semestre 1974. Laboratoire de Calcul des Probabilités, Université Paris VI (1974)Google Scholar
  14. 14.
    Maslov, V.P.: Théorie des perturbations et méthodes asymptotiques. Paris: Dunod 1972Google Scholar
  15. 15.
    Nelson, E.: Feynman integrals and the Schrödinger equation. J. Math. Phys.5, 332–343 (1964)Google Scholar
  16. 16.
    Schilder, M.: Some asymptotic formulas for Wiener integrals. Trans. Am. Math. Soc.125, 63–85 (1966)Google Scholar
  17. 17.
    Stroock, D.W., Varadhan, S.R.S.: On the support of diffusion processes with applications to the strong maximum principles. 6Lh Berkeley Symposium, Vol. III, 1972Google Scholar
  18. 18.
    Williams, D.: On a stopped Brownian motion formula of H.M. Taylor. Séminaire de Probabilité X. Lectures notes in mathematics511, pp. 235–239. Berlin, Heidelberg, New York: Springer 1976Google Scholar

Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • Halim Doss
    • 1
  1. 1.Laboratoire de ProbabilitésParis Cedex 05France

Personalised recommendations