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Bulletin of the American Mathematical Society

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Book Review

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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.

References [Enhancements On Off] (What's this?)

  • 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. 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.

Review Information:

Reviewer: Jerrold E. Marsden
Reviewer: Alan Weinstein
Journal: Bull. Amer. Math. Soc. 1 (1979), 545-553
DOI: https://doi.org/10.1090/S0273-0979-1979-14617-2
American Mathematical Society