Hostname: page-component-848d4c4894-m9kch Total loading time: 0 Render date: 2024-06-02T20:50:11.967Z Has data issue: false hasContentIssue false
  • English
  • Français

The stationary phase method with an estimate of the remainder term on a space of large dimension

Published online by Cambridge University Press:  22 January 2016

Daisuke Fujiwara
Daisuke Fujiwara*
Affiliation:
Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Oh-okayama, Meguro-ku Tokyo 152, Japan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In discussing convergence of Feynman path integrals [2], we need a stationary phase method of oscillatory integrals over a space of large dimension. More precisely, we have to know how the remainder term behaves when the dimension of the space goes to ∞ (cf. [2], [3] and [5]). The aim of the present note is to give answer to this question under rather mild assumptions. Application to the Feynman path integrals is discussed in [3] and [5].

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 124 , December 1991 , pp. 61 - 97
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1991

References

[ 1 ] Asada, K.-Fujiwara, D., On some oscillatory integral transformations in L2(Rn), Japan. J. Math., 4 (1978), 299–361.CrossRefGoogle Scholar
[ 2 ] Feynman, R. P., Space time approach to non relativistic quantum mechanics, Rev. Modern Phys., 20 (1948), 367–387.CrossRefGoogle Scholar
[ 3 ] Fujiwara, D., Remarks on convergence of some Feynman path integrals, Duke Math. J., 47 (1980), 559–600.CrossRefGoogle Scholar
[ 4 ] Fujiwara, D., A remark on Taniguchi-Kumanogo theorem for product of Fourier integral operators, Pseudo-differential operators, Proc. Oberwolfach 1986, Lecture notes in Math. No. 1256, Springer (1987), 135–153.Google Scholar
[ 5 ] Fujiwara, D., The Feynman path integral as an improper integral over the Sobolev space, Proc. of “Journee equations aux derivee partielles, Sain Jean de Monts 1990”, Société Math de France (1990), XIV 1–15.Google Scholar
[ 6 ] Kumanogo, H., Pseudo differential operators, MIT Press, (1982).Google Scholar
[ 7 ] Kumanogo, H.-Taniguchi, K., Fourier integral operators of multiphase and the fundamental solution for a hyperbolic system, Funkci. Ekvac, 22 (1979), pp. 161–196.Google Scholar
[ 8 ] Taniguchi, K., Multi-products of Fourier integral operators and the fundamental solution for a hyperbolic system with involutive characteristics, Osaka J. Math., 21 (1984),169–124.Google Scholar