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Oscillatory integrals and the method of stationary phase in infinitely many dimensions, with applications to the classical limit of quantum mechanics. I. (English) Zbl 0449.35092

MSC:
35Q99 Partial differential equations of mathematical physics and other areas of application
58J99 Partial differential equations on manifolds; differential operators
81Q30 Feynman integrals and graphs; applications of algebraic topology and algebraic geometry
35S99 Pseudodifferential operators and other generalizations of partial differential operators
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[1] Erdélyi, A.: Asymptotic expansions. Dover 1956 · Zbl 0070.29002
[2] Heading, J.: An introduction to phase-integral methods. London-New York: Wiley 1962 · Zbl 0115.07102
[3] Maslov, V.P.: The quasi-classical asymptotic solutions of some problems in mathematical physics I. J. Comp. Math.1, 123-141 (1961) (transl.); II. J. of Comp. Math.1, 744-778 (1961) (transl.)
[4] Maslov, V.P.: Théorie des perturbations et méthodes asymptotiques. Paris: Dunod 1972 (transl.)
[5] Maslov, V.P.: Characteristics of pseudo-differential operators. Proc. Internat. Congress Math. (Nice, 1970), Vol.2. Paris: Gauthiers-Villars 1971
[6] Leray, J.: Solutions asymptotiques et groupe symplectique. In: Fourier integral operators and partial differential equations, pp. 73-97. Lecture Notes in Math.459. Berlin-Heidelberg-New York: Springer 1975
[7] Hörmander, L.: Fourier integral operators I. Acta Math.127, 79-183 (1971) · Zbl 0212.46601 · doi:10.1007/BF02392052
[8] Duistermaat, J.J., Hörmander, L.: Fourier integral operators II. Acta Math.128, 183-269 (1972) · Zbl 0232.47055 · doi:10.1007/BF02392165
[9] Hörmander, L.: The calculus of Fourier integral operators. In: Prospects in mathematics. Ann. of Math. Studies70. Princeton: Princeton University Press 1971 · Zbl 0235.47023
[10] Hörmander, L.: On the existence and the regularity of solutions of linear pseudo-differential equations. L’enseignement Math.17, 99-163 (1971) · Zbl 0224.35084
[11] Duistermaat, J.J.: Fourier integral operators. Courant Institute Lecture Notes. New York, 1973 · Zbl 0272.47028
[12] Duistermaat, J.J.: Oscillatory integrals. Lagrange immersions and unfolding of singularities. Comm. Pure Appl. Math.27, 207-281 (1974) · Zbl 0285.35010 · doi:10.1002/cpa.3160270205
[13] Arnold, V.I.: Integrals of quickly oscillating functions and singularities of projections of Lagrange manifolds. Funct. Analys. and its Appl.6 (3), 222-224 (1972) (transl.) · Zbl 0278.57010 · doi:10.1007/BF01077879
[14] Arnold, V.I.: Remarks on the stationary phase method and Coxeter numbers. Russ. Math. Surveys28, 19-48 (1973) · Zbl 0291.40005 · doi:10.1070/RM1973v028n05ABEH001609
[15] Arnold, V.I.: Classification of unimodal critical points of functions. Funct. Analys. and its Appl.7 (3), 230-231 (1973) (transl.) and references given therein, see also e.g. [7, 8] · Zbl 0294.57018 · doi:10.1007/BF01080701
[16] Arnold, V.I.: Normal forms of functions in neighborhoods of degenerate critical points. Russ. Math. Surveys29 (2), 10-50 (1974) (transl.) · Zbl 0304.57018 · doi:10.1070/RM1974v029n02ABEH003846
[17] Bernshtein, I.N.: Modules over a ring of differential operators. Study of the fundamental solution of equations with constant coefficients. Funct. Analys. and its Appl.5 (2), 89-101 (1971) · Zbl 0233.47031 · doi:10.1007/BF01076413
[18] Malgrange, B.: Integrales asymptotiques et monodromie. Ann. Scient. Ec. Norm. Sup., 43 Série,7, 405-430 (1974) · Zbl 0305.32008
[19] Guillemin, V., Schaeffer, D.: Remarks on a paper of Ludwig. Bull. Am. Math. Soc.79, 382-385 (1973) · Zbl 0256.35008 · doi:10.1090/S0002-9904-1973-13176-3
[20] Thom, R.: Stabilité structurelle et morphogénèse. Reading: Benjamin 1972
[21] Albeverio, S., Høegh-Krohn, R.: Mathematical theory of Feynman path integrals. Lecture Notes in Math.523. Berlin-Heidelberg-New York: Springer 1976 · Zbl 0337.28009
[22] Berry, M.V., Mount, K.E.: Semiclassical approximations in wave mechanics. Rep. Progr. Phys.35, 315-397 (1972) · doi:10.1088/0034-4885/35/1/306
[23] Voros, A.: Semi-classical approximations. Ann. Inst. H. Poinc.24, 31-90 (1976)
[24] Fröman, N., Fröman, P.O.: JWKB approximation. Contributions to the theory. Amsterdam: North Holland 1965 · Zbl 0129.41907
[25] Birkhoff, G.D.: Quantum mechanics and asymptotic series. Bull. Am. Math. Soc.39, 681-700 (1933) · Zbl 0008.08902 · doi:10.1090/S0002-9904-1933-05716-6
[26] Keller, J.B., Lewis, R.M., Seckler, B.D.: Asymptotic solutions of some differential problems. Commun. Pure and Appl. Math.9, 207-265 (1956) · Zbl 0073.44105 · doi:10.1002/cpa.3160090205
[27] Lax, P.D.: Asymptotic solutions of oscillatory initial value problems. Duke Math. J.24, 627-646 (1957) · Zbl 0083.31801 · doi:10.1215/S0012-7094-57-02471-7
[28] Babi?, V.M., Ed.: Mathematical question in the theory of wave diffraction I, Proc. Stekl. Inst. Math.115 (1971) (transl.); II, Stekl. Math. Inst., Leningrad15 (1971) (transl.)
[29] Andrié, M.: Der klassische Grenzfall der Quantentheorie. Comment. Physico-Math.41, 333-351 (1971) · Zbl 0224.35012
[30] Hepp, K.: The classical limit for quantum mechanical correlation functions. Comm. math. Phys.35, 265-277 (1974) · doi:10.1007/BF01646348
[31] Hepp, K.: On the classical limit in quantum mechanics. Lecture given at the International School in Mathematical Physics, Camerino, 1974, to be published in Astérisque
[32] Lieb, E.H.: The classical limit of quantum spin systems. Commun. math. Phys.31, 327-340 (1973) · Zbl 1125.82305 · doi:10.1007/BF01646493
[33] Hepp, K., Lieb, E.H.: Constructive macroscopic quantum electrodynamics, in Constructive quantum field theory, G. Velo and A. Wightman, eds. Berlin-Heidelberg-New York: Springer 1973
[34] Albeverio, S., Høegh-Krohn, R.: Homogeneous random fields and statistical mechanics. J. Funct. Analys.19, 242-272 (1975) · Zbl 0381.60049 · doi:10.1016/0022-1236(75)90058-0
[35] Jost, R.: Poisson brackets (an unpedagogical lecture). Rev. Mod. Phys.36, 572-579 (1964) · doi:10.1103/RevModPhys.36.572
[36] Berezin, F.: Quantization, lectures given at XII Winter School in Karpacz, 1975, to be published
[37] Eckmann, J.P., Sénéor, R.: The Maslov-WKB method for the harmonic oscillator. Arch. Rat. Mech. Anal.61, 153-173 (1976) · Zbl 0332.34053 · doi:10.1007/BF00249703
[38] Colin de Verdière, Y.: Spectre du Laplacien et longueurs des géodésiques périodiques, I, II. Comp. Math.27, 83-106, 159-184 (1973) · Zbl 0272.53034
[39] Duistermaat, J.J. Guillemin, V.W.: The spectrum of positive elliptic operators and periodic bicharacteristic. Inventiones math.29, 39-79 (1975) · Zbl 0307.35071 · doi:10.1007/BF01405172
[40] Huber, A.: Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgruppen. Math. Ann.138, 1-26 (1959) · Zbl 0089.06101 · doi:10.1007/BF01369663
[41] Fierz, M.: Private communication
[42] Dirac, P.A.M.: On the analogy between classical and quantum mechanics. Rev. Mod. Phys.17, 195-199 (1945) · Zbl 0060.45102 · doi:10.1103/RevModPhys.17.195
[43] Feynman, R.P.: Space-time approach to non-relativistic quantum mechanics. Rev. Mod. Phys.20, 367-387 (1948) · Zbl 1371.81126 · doi:10.1103/RevModPhys.20.367
[44] Morette, C.: On the definition and approximation of Feynman’s path integrals. Phys. Rev.81, 848-852 (1951) · Zbl 0042.45506 · doi:10.1103/PhysRev.81.848
[45] Groenewold, H.J.: Quasi-classical path integrals, Kgl. Danske Videnskab. Selskab, Mat-fys. Medd.30 (19), 1-36 (1956) · Zbl 0075.21804
[46] Pauli, W.: Ausgewählte Kapitel aus der Feldquantisierung, ausg. U. Hochstrasser-M.F. Schafroth, E.T.H., Zürich (1951), Appendix
[47] Choquard, Ph.: Traitement semi-classique des forces générales dans la représentation de Feynman. Helv. Phys. Acta28, 89-157 (1955) · Zbl 0064.21603
[48] Maslov, V.P.: Stationary phase method for Feynman’s continual integral. Theor. and Math. Phys.2-3, 21-25 (1970) (transl.) · doi:10.1007/BF01028852
[49] Kato, T.: Perturbation theory for linear operators. New York: Springer 1966 · Zbl 0148.12601
[50] Arnold, V.I.: On a characteristic class which enters in quantization conditions. Funct. Analys. Appl.1, 1-13 (1967) (transl.) · doi:10.1007/BF01075861
[51] Daletskii, Yu.L.: Integration in function spaces. In: Progress in mathematics,4, R.V. Gamkrelidze, ed., pp. 87-132. transl. New York: Plenum 1969
[52] Colin de Verdière, Y.: Parametrix de l’équation des ondes et intégrales sur l’espace des chemins, Séminaire Goulaouic Lions-Schwartz Exp. XX, 1975
[53] Albeverio, S.: Lectures on the mathematical theory of Feynman path integrals, pp. 140-205. In: Proceedings of the conference on probabilistic and functional methods in quantum field theory (XII-th Winter School of Theoretical Physics, Karpacz, 1975). Acta Univ. Wratisl. No. 368, 1976
[54] Maslov, V.P., Chebotarev, A.M.: Generalized measures and Feynman path integrals. Teor. i Mat. Fyz.28, 3, 291-306 (1976) (Russ.) · Zbl 0351.28007
[55] Truman, A.: Feynman path integrals and quantum mechanics as ??0. J. Math. Phys.17, 1852-1862 (1976) · doi:10.1063/1.522806
[56] Combe, Ph., Rideau, G., Rodriguez, R., Sirugue-Collin, M.: On some mathematical problems in the definition of Feynman path integral. Marseille Preprint, July 1976 · Zbl 0421.28014
[57] Schulman, L.S.: Caustics and multivaluednes: two results of adding path amplitudes, pp. 144-156. In: Functional integration and its applications, A.M. Arthurs, ed. Oxford: Clarendon Press 1975 · Zbl 0336.46065
[58] Grossmann, A., Seiler, R.: Heat equation on phase space and the classical limit of quantum mechanical expectation values. Comm. Math. Phys.48, 195-197 (1976) · doi:10.1007/BF01617868
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