Semi-classical approximations

Voros, A.

Annales de l'institut Henri Poincaré. Section A, Physique Théorique, Tome 24 (1976) no. 1, pp. 31-90.

[1] A. Messiah, Mécanique Quantique, Dunod, Paris, 1965. | MR

[2] L. Landau et E. Lifschitz, Mécanique Quantique, Mir, Moscow.

[3] L. Landau et E. Lifschitz, Mécanique, Mir, Moscow.

[4] L. Janossy, Found of Physics, t. 3, 1973, p. 185. | MR

[5] J.B. Keller, Ann. Phys., t. 4, 1958, p. 180. | MR | Zbl

[6] D. Ludwig, Comm. Pure Appl. Math., t. 13, 1960, p. 473. | MR | Zbl

[7] J.B. Keller and S. Rubinow, Phys. Rev., t. 131, 1963, p. 2789. | MR

[8] J. Milnor, Morse Theory . Princeton Univ. Press, 1963. | MR | Zbl

[9] E. Wigner, Phys. Rev., t. 40, 1932, p. 749 ; K. Imre Et Al., J. Math. Phys., t. 8, 1967, p. 1097 ; A. Grossmann, G. Loupias and E.M. Stein, Ann. Inst. Fourier, t. 18, 1969, p. 343.

[10] V. Maslov, Théorie des Perturbations. Dunod, Paris, 1972 : this French translation (by J. Lascoux and R. Seneor) includes the articles of ref. [11]. | Zbl

[11] V. Arnold, Funct. Anal., t. 1, 1967, p. 1 ; V. Bouslaiev, Funct. Anal., t. 3, 1969, p. 17 (in Russian, see ref. [10]).

[12] J. Leray, Proc. « RCP 25 », vol. 18, Strasbourg University, publ. 1973.

[13] V. Arnold and A. Avez, Problèmes Ergodiques de la Mécanique Classique. Gauthier-Villars, Paris, 1967. | MR | Zbl

[14] H. Poincaré, Méthodes Nouvelles de la Mécanique Céleste. Gauthier-Villars, Paris, 1892; A.T. Whittaker, Treatise on Analytical Dynamics. Cambridge, Univ. Press, 1959.

[15] K. Hepp, Comm. Math. Phys., t. 35, 1974, p. 265. | MR

[16] R. Balian and C. Bloch, Ann. Phys., t. 60, 1970, p. 401 ; t. 63, 1971, p. 592; t. 64, 1971, p. 271 ; t. 69, 1972, p. 76. | Zbl

[17] R. Balian and C. Bloch, Ann. Phys., t. 85, 1974, p. 514. | MR | Zbl

[18] M.C. Gutzwiller, J. Math. Phys., t. 11, 1970, p. 1791 ; t. 12, 1971, p. 343, and references therein.

[19] R. Feynman and A. Hibbs, Quantum Mechanics and Path Integrals. McGraw-Hill, 1965. | Zbl

[20] I. Fenyes, Z. Phys., t. 132, 1952, p. 81 ; E. Nelson, Dynamical Theories of Brownian Motion. Princeton Univ. Press, 1972. | MR

[21] For instance: J. Heading, Introduction to Phase Integral Methods. Methuen, London, 1967 ; N. Fröman and P. Fröman, JWKB approximation. North Holland, 1965.

[22] M. Berry and K. Mount, Rep. Prog. Phys., t. 35, 1972, p. 315.

[23] R. Abraham and J. Marsden, Foundations of Mechanics. Benjamin, 1967.

[24] J.M. Souriau, Structure des Systèmes Dynamiques. Dunod, Paris, 1970. | MR | Zbl

[25] C. Godbillon, Géométrie Différentielle et Mécanique Analytique. Hermann, Paris, 1971. | MR

[26] R. Courant and D. Hilbert, Methods of Mathematical Physics. II. Interscience, 1962. | Zbl

[27] H. Weyl and H. Robertson, The Theory of Groups and Quantum Mechanics. Dover.

[28] L. Hörmander, L'Enseignement Mathématique, t. 17, 1971, p. 99 ; L. Hörmander, Acta Math., t. 127, 1971, p. 79 ; L. Hörmander and J. Duistermaat, Acta Math., t. 128, 1972, p. 183. | MR

[29] D. Iagolnitzer, Introduction to S-Matrix Theory. ADT, Paris, 1973.

[30] K. Hepp and E. Lieb, in Constructive Field Theory. Springer Lecture Notes in Physics, t. 25, 1973. | MR

[31] L. Schwartz, Distributions. Hermann, 1957 ; K. Yosida, Functional Analysis, chap. 1, Springer, 1968. | MR

[32] J.P. Eckmann and R. Seneor, The Maslov-WKB Method for the Harmonic Oscillator. To appear in Arch. Rational Mechanics. | MR

[33] For instance: H. Abarbanel, Cargèse Lectures , vol. 5. Gordon and Breach, 1972.

[34] A.I. Khinchin, Mathematical Foundations of Statistical Mechanics. Dover, 1949. | MR | Zbl

[35] I.E. Farquhar, Ergodic Theory in Statistical Mechanics. Interscience, 1964. | MR

[36] V. Arnold, Funct. Anal. and Applications (Russian), t. 6 (II), 1972, p. 12. | Zbl

[37] V. Lazutkin, Izv. Akad. Nauk USSR (Math.), t. 37, 1973, p. 186 and 437 and references therein. | Zbl

[38] J.B. Keller and S. Rubinov, Ann. Phys., t. 9, 1960, p. 24. | Zbl

[39] S. Sternberg, Celestial Mechanics. II. Benjamin, 1969. | Zbl

[40] A. Voros, Proc. « RCP 25 », vol. 18. Strasbourg University, publ. 1973.

[41] A.J. Hanson and T. Regge, IAS. Princeton preprint, 1974. | MR

[42] B. Kostant, Springer Lecture Notes in Mathematics, t. 170, 1970 ; P. Renouard, Thesis, Orsay, 1969. | MR

[43] For instance: Proc. of the 1974 Conference of Mathematical Physics. Warsaw, to be published ; E. ONOFRI et M. PAURI, J. Math. Phys., t. 13, 1972, p. 533 and references therein.

[44] Y. Colin De Verdière, Compositio Math., t. 3, 1973 ; J. Chazarain, C. R. Acad. Sci. Paris, t. 277, 1973, p. 595 ; J. Duistermaat and V. Guillemin, Proc. Symp. Diff. Geometry, August 1973, Stanford.

[45] T. Kato, Perturbation Theory. Springer, 1966, p. 443. | Zbl

[46] For instance: Y. Iwasaki, Prog. Theor. Phys., t. 46, 1971, p. 1587.

[47] I.M. Gelfand and F. Chilov, Generalized Functions. Academic Press, 1968. | Zbl

[48] For instance, J. Iliopoulos, C. Itzykson and A. Martin, Saclay preprint DPh-T/74-18.

[49] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions. Dover, 1970.

[50] I.C. Percival, J. Phys. (Letters), B 6, 1973, L229.

[51] G.W. Mackey, Mathematical Foundations of Quantum Mechanics. Benjamin, 1963. | Zbl

[52] I.M. Gelfand and V.B. Lidski, Usp. Mat. Nauk, t. 10-1, 1955, p. 3 and A. M. S. Translations, t. 8, 1958, p. 143. | Zbl

[53] A. Voros, Proc. « Colloque Symplectique ». Aix-en-Provence, June 1974, to be published by the C. N. R. S., Paris.

[54] A. Voros, Proc. « RCP 25 » of May 1975, Strasbourg University (to appear).