Advertisement

Communications in Mathematical Physics

, Volume 101, Issue 2, pp 187–206 | Cite as

The method of stationary phase for oscillatory integrals on Hilbert spaces

  • Jorge Rezende
Article

Abstract

We develop the method of stationary phase for the normalized-oscillatory integral on Hilbert space, giving Borel summable expansions. The developments that we obtain hold for more general situations than the ones of previous papers on the same subject.

Keywords

Neural Network Statistical Physic Hilbert Space Stationary Phase Complex System 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Albeverio, S., Høegh-Krohn, R.: Mathematical theory of Feynman Path integrals. Lecture Notes in Mathematics, Vol.523. Berlin, Heidelberg, New York: Springer 1976Google Scholar
  2. 2.
    Albeverio, S., Høegh-Krohn, R.: Oscillatory integrals and the method of stationary phase in infinitely many dimensions, with applications to the classical limit of quantum mechanics I. Invent. Math.40, 59–106 (1977)Google Scholar
  3. 3.
    Albeverio, S., Høegh-Krohn, R.: Feynman Path integrals and the corresponding method of stationary phase. Lecture Notes in Physics, Vol.106, pp. 3–57. Berlin, Heidelberg, New York: Springer 1979Google Scholar
  4. 4.
    Albeverio, S., Blanchard, Ph., Høegh-Krohn, R.: Feynman Path integrals and the trace formula for the Schrödinger operators. Commun. Math. Phys.83, 49–76 (1982)Google Scholar
  5. 5.
    Rezende, J.: Remark on the solution of the Schrödinger equation for anharmonic oscillators via the Feynman Path integral. Lett. Math. Phys.7, 75–83 (1983)Google Scholar
  6. 6.
    Truman, A.: The polygonal Path formulation of the Feynman Path integral. Lecture Notes in Physics, Vol.106, pp. 73–102. Berlin, Heidelberg, New York: Springer 1979Google Scholar
  7. 7.
    Elworthy, D., Truman, A.: Feynman maps, Cameron-Martin formulae and an harmonic oscillators. Ann. Inst. Henri Poincare41, 115–142 (1984)Google Scholar
  8. 8.
    Hörmander, L.: Fourier integral operators I. Acta Math.127, 79–183 (1971)Google Scholar
  9. 9.
    Fedoriuk, M. V.: The stationary phase method and pseudodifferential operators. Russ. Math. Surv.26, 65–115 (1971)Google Scholar
  10. 10.
    Maslov, V. P., Fedoriuk, M. V.: Semi-classical approximation in quantum mechanics. Dordrecht: D. Reidel Publishing Company 1981Google Scholar
  11. 11.
    Schilder, M.: Some asymptotic formulas for Wiener integrals. Trans. Am. Math. Soc.125, 63–85 (1966)Google Scholar
  12. 12.
    Watson, G. N.: An expansion related to Stirling's formula, derived by the method of steepest descents. Q. J. Pure. Appl. Math.48, 1–18 (1920)Google Scholar
  13. 13.
    Riordan, J.: Combinatorial identities. New York: John Wiley 1968Google Scholar
  14. 14.
    Nevanlinna, F.: Zur Theorie der asymptotischen Potenzreihen. Ann. Acad. Sci. Fenn. (A)12, no. 3 (1976)Google Scholar
  15. 15.
    Bieberbach, L.: Jahrb, Forfschr. Math.46, 1463–1465 (1916–1918)Google Scholar
  16. 16.
    Simon, B.: Large orders and summability of eigenvalue perturbation theory: A mathematical overview. Int. J. Quant. Chem.21, 3–25 (1982)Google Scholar
  17. 17.
    Poincaré, H.: Sur les intégrales irrégulières des equations linéaires. Acta. Math.8, 295–344 (1886)Google Scholar
  18. 18.
    Borel, E.: Oeuvres, pp. 399–568. Paris: C.N.R.S. 1972Google Scholar
  19. 19.
    Borel, E.: Mémoire sur les séries divergentes. Ann. Ec. Norm. Sup. 3e.série,t.16, 9–131 (1899)Google Scholar
  20. 20.
    Watson, G. N.: The transformation of an asymptotic series into a convergent series of inverse factorials. Rend. Circ. Mat. Palermo34, 41–88 (1912)Google Scholar
  21. 21.
    Watson, G. N.: A theory of asymptotic series. Trans. Royal Soc. London A211, 279–313 (1912)Google Scholar
  22. 22.
    Hardy, G. H.: Divergent series. London: Oxford University Press 1963Google Scholar
  23. 23.
    Sokal, A.: An improvement of Watson's theorem on Borel summability. J. Math. Phys.21, 261–263 (1980)Google Scholar
  24. 24.
    Dieudonné, J.: Calcul infinitésimal. Paris: Hermann 1968Google Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • Jorge Rezende
    • 1
  1. 1.Fakultät für PhysikUniversität BielefeldBielefeld 1Federal Republic of Germany

Personalised recommendations