Bulletin of the American Mathematical Society

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Book Information:

Authors: Victor Guillemin and Shlomo Sternberg
Title: Geometric asymptotics
Additional book information: Math. Surveys, no. 14, Amer. Math. Soc., Providence, R.I., 1977, xviii + 474 pp., $34.40.

Author: Nolan R. Wallach
Title: Symplectic geometry and Fourier analysis
Additional book information: Math Sci. Press, Brookline, Massachusetts, 1977, xvii + 436 pp.

R. Abraham [1967], Foundations of mechanics, Benjamin, New York.

Ralph Abraham and Jerrold E. Marsden, Foundations of mechanics, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978. Second edition, revised and enlarged; With the assistance of Tudor Raţiu and Richard Cushman. MR 515141

V. I. Arnol′d, On a characteristic class entering into conditions of quantization, Funkcional. Anal. i Priložen. 1 (1967), 1–14 (Russian). MR 0211415

V. Arnold, Les méthodes mathématiques de la mécanique classique, Éditions Mir, Moscow, 1976 (French). Traduit du russe par Djilali Embarek. MR 0474391

G. D. Birkhoff [1931], Quantum mechanics and asymptotic series. I, Bull. Amer. Math. Soc. 39, 681-700.

Paul R. Chernoff and Jerrold E. Marsden, Properties of infinite dimensional Hamiltonian systems, Lecture Notes in Mathematics, Vol. 425, Springer-Verlag, Berlin-New York, 1974. MR 0650113

J. J. Duistermaat, Oscillatory integrals, Lagrange immersions and unfolding of singularities, Comm. Pure Appl. Math. 27 (1974), 207–281. MR 405513, DOI 10.1002/cpa.3160270205

J. J. Duistermaat and L. Hörmander, Fourier integral operators. II, Acta Math. 128 (1972), no. 3-4, 183–269. MR 388464, DOI 10.1007/BF02392165

Yu. V. Egorov [1969], On canonical transformations of pseudo-differential operators, Uspehi Mat. Nauk 25, 235-236.

Claude Godbillon, Géométrie différentielle et mécanique analytique, Hermann, Paris, 1969 (French). MR 0242081

Robert Hermann, Differential geometry and the calculus of variations, 2nd ed., Interdisciplinary Mathematics, Vol. XVII, Math Sci Press, Brookline, Mass., 1977. MR 0494184

Lars Hörmander, Fourier integral operators. I, Acta Math. 127 (1971), no. 1-2, 79–183. MR 388463, DOI 10.1007/BF02392052

Joseph B. Keller, Corrected Bohr-Sommerfeld quantum conditions for nonseparable systems, Ann. Physics 4 (1958), 180–188. MR 99207, DOI 10.1016/0003-4916(58)90032-0

E. C. Kemble [1937], The fundamental principles of quantum mechanics, McGraw-Hill, New York.

A. A. Kirillov, Unitary representations of nilpotent Lie groups, Uspehi Mat. Nauk 17 (1962), no. 4 (106), 57–110 (Russian). MR 0142001

Bertram Kostant, Quantization and unitary representations. I. Prequantization, Lectures in modern analysis and applications, III, Lecture Notes in Math., Vol. 170, Springer, Berlin, 1970, pp. 87–208. MR 0294568

Jean Leray, Analyse lagrangienne et mécanique quantique, Séminaire sur les Équations aux Dérivées Partielles (1976–1977), I, Exp. No. 1, Collège de France, Paris, 1977, pp. 303 (French). MR 0501198

G. W. Mackey [1963], The mathematical foundations of quantum mechanics, Benjamin, New York.

J. E. Marsden, Hamiltonian one parameter groups: A mathematical exposition of infinite dimensional Hamiltonian systems with applications in classical and quantum mechanics, Arch. Rational Mech. Anal. 28 (1967/68), 362–396. MR 226145, DOI 10.1007/BF00251662

Jerrold Marsden and Alan Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Mathematical Phys. 5 (1974), no. 1, 121–130. MR 402819, DOI 10.1016/0034-4877(74)90021-4

V. P. Maslov [1965], Theory of perturbations and asymptotic methods, Moscow State Univ. (French translation, Dunod, 1972).

V. P. Maslov and M. V. Fedoryuk, Kvaziklassicheskoe priblizhenie dlya uravneniĭ kvantovoĭ mekhaniki, Izdat. “Nauka”, Moscow, 1976 (Russian). MR 0461590

S. C. Miller Jr. and R. H. Good Jr., A WKB-type approximation to the Schrödinger equation, Phys. Rev. (2) 91 (1953), 174–179. MR 59638

I. E. Segal, Quantization of nonlinear systems, J. Mathematical Phys. 1 (1960), 468–488. MR 135093, DOI 10.1063/1.1703683

I. Segal [1965], Differential operators in the manifold of solutions of a nonlinear differential equation, J. Math. Pures Appl. 44, 71-113.

J. J. Sławianowski, Quantum relations remaining valid on the classical level, Rep. Mathematical Phys. 2 (1971), no. 1, 11–34. MR 285224, DOI 10.1016/0034-4877(71)90016-4

J.-M. Souriau, Structure des systèmes dynamiques, Dunod, Paris, 1970 (French). Maîtrises de mathématiques. MR 0260238

Shlomo Sternberg, Lectures on differential geometry, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0193578

Michael Taylor, Pseudo differential operators, Lecture Notes in Mathematics, Vol. 416, Springer-Verlag, Berlin-New York, 1974. MR 0442523

W. M. Tulczyjew, The Legendre transformation, Ann. Inst. H. Poincaré Sect. A (N.S.) 27 (1977), no. 1, 101–114 (English, with French summary). MR 494271

Léon Van Hove, Sur certaines représentations unitaires d’un groupe infini de transformations, Acad. Roy. Belg. Cl. Sci. Mém. Coll. in 8$^\circ$ 26 (1951), no. 6, 102 (French). MR 57260

Alan Weinstein, Lectures on symplectic manifolds, Regional Conference Series in Mathematics, No. 29, American Mathematical Society, Providence, R.I., 1977. Expository lectures from the CBMS Regional Conference held at the University of North Carolina, March 8–12, 1976. MR 0464312

H. Weyl [1931], The theory of groups and quantum mechanics, Dover, New York.

R. Abraham [1967], Foundations of mechanics, Benjamin, New York.




R. Abraham and J. Marsden [1978], Foundations of mechanics, 2nd ed., Addison-Wesley, Reading, Mass. MR 0515141




V. Arnold [1967], Characteristic class entering in conditions of quantization, Functional Anal. Appl. 1, 1-13. MR 211415




V. Arnold, [1978], Mathematical methods of classical mechanics, Graduate Texts in Mathematics, vol. 60, Springer-Verlag, Berlin and New York. MR 690288




G. D. Birkhoff [1931], Quantum mechanics and asymptotic series. I, Bull. Amer. Math. Soc. 39, 681-700.




P. R. Chernoff and J. E. Marsden [1974], Properties of Infinite Dimensional Hamiltonian Systems, Lecture Notes in Math., vol. 425, Springer-Verlag, New York. MR 650113




J. J. Duistermaat [1974], Oscillatory integrals, Lagrange immersions and the unfolding of singularities, Comm. Pure Appl. Math. 27, 207-281. MR 405513




J. J. Duistermaat and L. Hörmander [1972], Fourier integral operators. II, Acta. Math. 128, 183-269. MR 388464




Yu. V. Egorov [1969], On canonical transformations of pseudo-differential operators, Uspehi Mat. Nauk 25, 235-236.




G. Godbillon [1969], Geometrie differentielle et mécanique analytique, Hermann, Paris. MR 242081




R. Hermann [1968], Differential geometry and the calculus of variations, Academic Press, New York (Second edition, [1977], Math. Sci. Press). MR 494184




L. Hörmander [1971], Fourier integral operators. I, Acta. Math. 127, 79-183. MR 388463




J. B. Keller [1958], Corrected Bohr-Sommerfeld quantum conditions for nonseparable systems, Ann. of Physics 4, 180-188. MR 99207




E. C. Kemble [1937], The fundamental principles of quantum mechanics, McGraw-Hill, New York.




A. A. Kirillov [1962], Unitary representations of nilpotent Lie groups, Russian Math. Surveys 17, 53-104. MR 142001




B. Kostant [1970], Quantization and unitary representations, Lecture Notes in Math., vol. 170, Springer-Verlag, Berlin and New York, pp. 87-208. MR 294568




J. Leray [1978], Analyse lagrangienne et mécanique quantique, R.C.P. 25, vol. 25, I.R.M.A. Strasbourg. MR 501198




G. W. Mackey [1963], The mathematical foundations of quantum mechanics, Benjamin, New York.




J. E. Marsden [1968], Hamiltonian one parameter groups, Arch. Rational Mech. Anal. 28, 362-396. MR 226145




J. Marsden and A. Weinstein [1974], Reduction of symplectic manifolds with symmetry, Rep. Mathematical Phys. 5, 121-130. MR 402819




V. P. Maslov [1965], Theory of perturbations and asymptotic methods, Moscow State Univ. (French translation, Dunod, 1972).




V. P. Maslov and M. V. Fedorjuk [1976], Quasiclassical approximation for the equations of quantum mechanics, Izdat. "Nauka", Moscow (Russian) MR 461590




S. C. Miller, Jr. and R. H. Good, Jr. [1953], A WKB-type approximation to the Schrödinger equation, Phys. Rev. 91, 174-179. MR 59638




I. Segal [1960], Quantization of non-linear systems, J. Math. Phys. 1, 468-488. MR 135093




I. Segal [1965], Differential operators in the manifold of solutions of a nonlinear differential equation, J. Math. Pures Appl. 44, 71-113.




J. J. Słwianowski [1971], Quantum relations remaining valid on the classical level, Rep. Mathematical Phys. 2, 11-34. MR 285224




J. M. Souriau [1970], Structure des systemes dynamiques, Dunod, Paris (2nd ed. in preparation). MR 260238




S. Sternberg [1964], Lectures on differential geometry, Prentice-Hall, Englewood Cliffs, N. J. MR 193578




M. Taylor [1979], Pseudo differential operators, (to appear). MR 442523




W. Tulczyjew [1977], The Legendre transformation, Ann. Inst. H. Poincaré, 27, 101-114. MR 494271




L. Van Hove [1951], Sur certaines représentations unitaires d'un groupe infini de transformations, Acad. Roy. Belg. Cl. Sci. Mem. Coll. in-8^° 26, pp. 61-102. MR 57260




A. Weinstein [1977], Lectures on symplectic manifolds, CBMS Regional Conf. Ser. in Math., no. 29, Amer. Math. Soc., Providence, R. I. MR 464312




H. Weyl [1931], The theory of groups and quantum mechanics, Dover, New York.