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Remarks on Fujiwara’s stationary phase method on a space of large dimension with a phase function involving electromagnetic fields

Published online by Cambridge University Press:  22 January 2016

Tetsuo Tsuchida
Tetsuo Tsuchida*
Affiliation:
Department of Mathematics, Faculty of Science, Kanazawa University, 920-11 Kanazawa, Japan
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We consider an oscillatory integral of the form

Here each xj, j = 0, 1,…, L, runs in Rd, ν > 1 is a constant and tj, j = 1,…,L, are positive constants. Fujiwara [5] discussed this integral for L large and developed the stationary phase method with an estimate of the remainder term for the phase function S(xL,…, x0) coming from the action integral for a particle in an electric field. But his results cannot be applied to the integral which naturally arises in the discussion of quantum mechanics of a charged particle moving in a magnetic field. In this paper we extend his results to the case for the phase function involving both electric and magnetic fields.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 136 , December 1994 , pp. 157 - 189
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1994

References

[1] Asada, K., Fujiwara, D., On some oscillatory integral transformations in L2 (R n ), 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), 367–386.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., 1256, Springer, 135–153 (1987).Google Scholar
[5] Fujiwara, D., The stationary phase method with an estimate of the remainder term on a space of large dimension, Nagoya Math. J., 124 (1991), 61–97.CrossRefGoogle Scholar
[6] Fujiwara, D., Some Feynman path integrals as oscillatory integrals over a Sobolev manifold, Lecture Notes in Math., 1540, Springer (1993), 39–53.Google Scholar
[7] Kumanogo, H., Pseudo-differential Operators, MIT Press, 1982.Google Scholar
[8] Nicoleau, F., Approximation semi-classique du propagateur d’un système électromagnétique et phénomène de Aharonov-Bohm, Helv. Phys. Acta., 65 (1992), 722–747.Google Scholar
[9] Yajima, K., Schrödinger evolution equations with magnetic fields, J. Analyse Math., 56 (1991), 29–76.CrossRefGoogle Scholar