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.
Similar content being viewed by others
References
Albeverio, S., Høegh-Krohn, R.: Mathematical theory of Feynman Path integrals. Lecture Notes in Mathematics, Vol.523. Berlin, Heidelberg, New York: Springer 1976
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)
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 1979
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)
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)
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 1979
Elworthy, D., Truman, A.: Feynman maps, Cameron-Martin formulae and an harmonic oscillators. Ann. Inst. Henri Poincare41, 115–142 (1984)
Hörmander, L.: Fourier integral operators I. Acta Math.127, 79–183 (1971)
Fedoriuk, M. V.: The stationary phase method and pseudodifferential operators. Russ. Math. Surv.26, 65–115 (1971)
Maslov, V. P., Fedoriuk, M. V.: Semi-classical approximation in quantum mechanics. Dordrecht: D. Reidel Publishing Company 1981
Schilder, M.: Some asymptotic formulas for Wiener integrals. Trans. Am. Math. Soc.125, 63–85 (1966)
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)
Riordan, J.: Combinatorial identities. New York: John Wiley 1968
Nevanlinna, F.: Zur Theorie der asymptotischen Potenzreihen. Ann. Acad. Sci. Fenn. (A)12, no. 3 (1976)
Bieberbach, L.: Jahrb, Forfschr. Math.46, 1463–1465 (1916–1918)
Simon, B.: Large orders and summability of eigenvalue perturbation theory: A mathematical overview. Int. J. Quant. Chem.21, 3–25 (1982)
Poincaré, H.: Sur les intégrales irrégulières des equations linéaires. Acta. Math.8, 295–344 (1886)
Borel, E.: Oeuvres, pp. 399–568. Paris: C.N.R.S. 1972
Borel, E.: Mémoire sur les séries divergentes. Ann. Ec. Norm. Sup. 3e.série,t.16, 9–131 (1899)
Watson, G. N.: The transformation of an asymptotic series into a convergent series of inverse factorials. Rend. Circ. Mat. Palermo34, 41–88 (1912)
Watson, G. N.: A theory of asymptotic series. Trans. Royal Soc. London A211, 279–313 (1912)
Hardy, G. H.: Divergent series. London: Oxford University Press 1963
Sokal, A.: An improvement of Watson's theorem on Borel summability. J. Math. Phys.21, 261–263 (1980)
Dieudonné, J.: Calcul infinitésimal. Paris: Hermann 1968
Author information
Authors and Affiliations
Additional information
Communicated by H. Araki
Supported by Deutscher Akademischer Austauschdienst (DAAD)
Rights and permissions
About this article
Cite this article
Rezende, J. The method of stationary phase for oscillatory integrals on Hilbert spaces. Commun.Math. Phys. 101, 187–206 (1985). https://doi.org/10.1007/BF01218758
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01218758