Asymptotic -expansions of stationary quantum states
Annales de l'I.H.P. Physique théorique, Tome 26 (1977) no. 4, p. 343-403
@article{AIHPA_1977__26_4_343_0,
     author = {Voros, A.},
     title = {Asymptotic $\hbar $-expansions of stationary quantum states},
     journal = {Annales de l'I.H.P. Physique th\'eorique},
     publisher = {Gauthier-Villars},
     volume = {26},
     number = {4},
     year = {1977},
     pages = {343-403},
     mrnumber = {456138},
     language = {en},
     url = {http://www.numdam.org/item/AIHPA_1977__26_4_343_0}
}
Voros, A. Asymptotic $\hbar $-expansions of stationary quantum states. Annales de l'I.H.P. Physique théorique, Tome 26 (1977) no. 4, pp. 343-403. http://www.numdam.org/item/AIHPA_1977__26_4_343_0/

[1] A. Messiah, Mécanique Quantique. Dunod, Paris, 1965. | MR 129304

[2] L. Landau-E. Lifschitz, Mécanique Quantique, Mir (Moscow).

[3] J.B. Keller, Ann. Phys., t. 4, 1958, p. 180. | MR 99207 | Zbl 0085.43103

[4] V.P. Maslov, Théorie des Perturbations. Dunod, Paris, 1972 (translated by J. LASCOUX and R. SÉNÉOR, includes refs. [45] [46]).

[5] R. Courant, D. Hilbert, Methods of Mathematical Physics II. Interscience, 1962. | Zbl 0099.29504

[6] D. Ludwig, Comm. Pure Appl. Math., t. 13, 1960, p. 473. | MR 115010 | Zbl 0098.29601

[7] L. Hörmander, Acta Math., t. 127, 1971, p. 79; L. Hormander, J.J. Duistermaat, Acta Math., t. 128, 1972, p. 183. | MR 388463 | Zbl 0232.47055

[8] J.J. Duistermaat, Fourier Integral Operators. Courant Institute Press, 1973. | MR 451313 | Zbl 0272.47028

[9] J. Chazarain (Ed.), Fourier Integral Operators and Partial Differential Equations (Proc. 1974, Conference, Nice). Springer Lecture Notes in Mathematics, vol. 459, 1975. | MR 388171 | Zbl 0299.00021

[10] H. Komatsu (Ed.), Hyperfunctions and Pseudo-differential Equations. Springer Lecture Notes in Mathematics, vol. 287, 1973 ; F. Pham (Ed.), Hyperfunctions and | MR 388082 | Zbl 0253.00006

Theoretical Physics. Springer Lectures Notes in Mathematics, vol. 449, 1975 ; J. Bros, D. Iagolnitzer, Ann. Inst. H. Poincaré, t. 18, 1973, p. 147 and Séminaire Goulaouic-Lions-Schwartz, Lecture n° 18, 1975. | Numdam

[11] Colloque International C. N. R. S., n° 237: Géométrie Symplectique et Physique Mathématique (Proc. 1974 Conference, Aix-en-Provence), C. N. R. S., Paris, 1975. | MR 424458

[12] J.J. Duistermaat, Comm. Pure Appl. Math., t. 27, 1974, p. 207. | MR 405513 | Zbl 0285.35010

[13] V. Guillemin, S. Sternberg, Geometric Asymptotics. A. M. S., to appear. | Zbl 0364.53011

[14] E. Wigner, Phys. Rev., t. 40, 1932, p. 749; J.C.T. Pool, J. Math. Phys., t. 7, 1966, p. 66; K. Imre et al., J. Math. Phys., t. 8, 1967, p. 1097.

[15] A. Grossmann, G. Loupias, E.M. Stein, Ann. Inst. Fourier, t. 18, 1969, p. 343. | Numdam | MR 267425 | Zbl 0176.45102

[16] H.J. Groenewold, Physica, t. 12, 1946, p. 405 ; J.E. Moyal, Proc. Cambridge Phil. Soc., t. 45, 1949, p. 99; R. Hermann, The Poisson-Moyal Bracket. Harvard University, Physics Dept. preprint, 1976. | MR 18562

[17] D. Kastler, Comm. Math. Phys., t. 1, 1965, p. 14 ; G. Loupias, S. Miracle-Sole, Comm. Math. Phys., t. 2, 1966, p. 31. | MR 193983 | Zbl 0137.45601

[18] N. March, J. Plaskett, Proc. Roy. Soc., t. A 235, 1956, p. 419. | MR 78500 | Zbl 0071.22601

[19] G. Baraff, S. Borowitz, Phys. Rev., t. 121, 1961, p. 1704 ; R. Balian, C. Bloch, Ann. Phys., t. 63, 1971, p. 592. | MR 118376

[20] R. Feynman, A. Hibbs, Quantum Mechanics and Path. Integrals. McGraw Hill, 1965; C. De Witt, Comm. Math. Phys., t. 28, 1972, p. 47 and t. 37, 1974, p. 63. | Zbl 0287.28006

[21] R. Balian, C. Bloch, Ann. Phys., t. 60, 1970, p. 401 ; t. 64, 1971, p. 271 ; t. 69, 1972, p. 76. | Zbl 0207.40202

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

[23] J. Leray, in [9] [11].

[24] J.M. Souriau, Comm. Math. Phys., t. 1, 1966, p. 374 and Structure des Systèmes Dynamiques. Dunod, Paris, 1970. | MR 207332

[25] B. Kostant, in Springer Lecture Notes in Mathematics, vol. 170, 1970. | MR 294568

[26] K. Maurin, R. Raczka, Mathematical Physics and Physical Mathematics. Proc., 1974, Conference, Warsaw, Reidel, 1976. | MR 550910 | Zbl 0327.00001

[27] A. Voros, Ann. Inst. H. Poincaré, t. 24, 1976, p. 31. | Numdam | MR 420747

[28] Y. Colin De Verdière, Compositio Math., t. 27, 1973, p. 159. | Numdam | MR 348798 | Zbl 0281.53036

[29] Y. Colin De Verdière, Quasi-Modes sur les Variétés Riemanniennes. Preprint, Paris VII University. | MR 501196

[30] J.J. Duistermaat, V. Guillemin, Inventiones Math., t. 29, 1975, p. 39. | MR 405514 | Zbl 0307.35071

[31] M. Gutzwiller, J. Math. Phys., t. 12, 1971, p. 343 ; A. Voros, in [11]; R. Dashen, B. Hasslacher, A. Neveu, Phys. Rev., t. D 10, 1974, p. 4114 and t. D 11, 1975, p. 3424.

[32] V. Guillemin, in [11].

[33] B. Kostant, Symposia Math., t. 14, 1974, p. 139; R. Blattner, B. Kostant, S. Stern- BERG, in [11].

[34] L. Van Hove, Acad. Roy. Belg. Bull. Class. Sci., t. 37, 1951, p. 610. | MR 45059 | Zbl 0044.23103

[35] I. Segal, Mat. Fys. Medd. Danske Vid. Selk., t. 31, 1959, p. 1; D. Shale, Trans. A. M. S., t. 103, 1962, p. 149; A. Weil, Acta Math., t. 111, 1963, p. 143. | MR 112626

[36] L. Landau, E. Lifschitz, Mécanique, Mir. Moscow.

[37] R. Abraham, J. Marsden, Foundations of Mechanics. Benjamin, 1967. | Zbl 0158.42901

[38] G. Mackey, Mathematical Foundations of Quantum Mechanics. Benjamin, 1967.

[39] L. Schwartz, Théorie des Distributions. Hermann, 1957. | MR 209834 | Zbl 0078.11003

[40] B. Simon, M. Reed, Methods of Modern Mathematical Physics. Academic Press, 1972 and 1975. | MR 493419

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

[42] V.V. Grushin, Funct. Anal. and Appl. (Russian), t. 4 (3), 1970, p. 37; L. Boutet De Monvel, Comm. Pure Appl. Math., t. 27, 1974, p. 585. | Zbl 0223.35084

[43] C. Rockland, Hypoellipticity and Eigenvalue Asymptotics. Lecture Notes in Mathematics. Springer, 1975, p. 464. | MR 501202 | Zbl 0302.35002

[44] H. Araki, Private communication.

[45] V. Arnold, Funct. Anal. and Appl. (russe), t. 1, 1967, p. 1 ; translated in [4].

[46] V. Bouslaiev, Funct. Anal. and Appl. (russe), t. 3, 1969, p. 17; translated in [4].

[47] M. Moshinsky, C. Quesne, J. Math. Phys., t. 12, 1971, p. 1772. | MR 286385 | Zbl 0247.20051

[48] G. Mackey, Induced Representations of Groups and Quantum Mechanics. Benjamin, 1968. | MR 507212 | Zbl 0174.28101

[49] J.M. Souriau, In 4th Colloquium on Group Theoretical Methods in Physics. Univ. Nijmegen, 1975.

[50] I.M. Gelfand, N. Vilenkin, Generalized Functions IV. Academic Press, 1964.

[51] K. Nomizu, Lie Groups and Differential Geometry. Math. Soc. Japan, 1956. | MR 84166 | Zbl 0071.15402

[52] N.N. Bogoliubov, J. E. T. P., t, 34, 1958, p. 58.

[53] F.A. Berezin, The Method of Second Quantization. Acad. Press, 1966. | MR 208930 | Zbl 0151.44001

[54] A. Weinstein, Bull. A. M. S., t. 82, 1976, p. 548. | MR 407595 | Zbl 0331.46034

[55] K.C. Liu, J. Math. Phys., t. 16, 1975, p. 724 and 2054; t. 17, 1976, p. 859. | Zbl 0296.46076

[56] J.P. Eckmann, R. Seneor, Arch. Rational Mechanics, t. 61, 1976, p. 153. | MR 406147 | Zbl 0332.34053

[57] V. Arnold, Funct. Anal. and Appl. (russe), t. 6 (II), 1972, p. 12. | Zbl 0251.70012

[58] A. Voros, Comptes Rendus « RCP 25 », vol. 22. Univ., Strasbourg, 1975.

[59] V. Arnold, A. Avez, Problèmes Ergodiques de la Mécanique Classique. Gauthier-Villars, 1967. | MR 209436 | Zbl 0149.21704

[60] A.C. Hearn, Reduce User's Manual. Second Edition, University of Utah, 1973.

[61] A. Voros, An Algebra of Pseudo-differential Operators..., J. Funct. Anal. (to appear). | MR 496088 | Zbl 0386.47031

[62] A. Voros, Ph. D. Thesis, Université Paris-Sud (Orsay), May 1977.