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Lagrangian manifolds and the Maslov operator. Transl. from the Russian by Dana Mackenzie. (English) Zbl 0727.58001
Springer Series in Soviet Mathematics. Berlin etc.: Springer-Verlag. x, 395 p. (1990).
The book is devoted to the theory of Maslov’s canonical operator and its applications to the quasi-classical asymptotics. It is self-contained and may be used as the first introduction to the subject. The exposition follows the lines of Maslov’s approach to the theory of global oscillating integrals.
The first part of the book contains some introductory material on the topology and geometry of real as well as complex Lagrangian manifolds. In particular, the Maslov index of a Lagrangian manifold is introduced.
The construction of Maslov’s canonical operators associated to a real Lagrangian manifold is presented in the second part. The development of the theory starts with the concept of local canonical operators, defined in any local chart of the Lagrangian manifold. If the Lagrangian manifold is quantized, then the local canonical operators coincide on the intersection of charts, giving rise of a Maslov’s canonical operator. A commutation formula between 1/h-pseudodifferential operators and Maslov canonical operators is proved. Main emphasis lies on the complex case. The Lagrangian manifold is s-analytic (almost analytic) in this case.
As an application, a construction of global asymptotic solutions of the Cauchy problem with highly oscillating initial data is presented. Asymptotics of the spectrum of 1/h-pseudodifferential operators are given as well. The relationship between the Maslov canonical operators and the theory of Fourier integral operators developed by L. Hörmander (real case) and by A. Melin, J. Sjostrand (complex case) is discussed in the appendix.
Reviewer: G.Popov (Sofia)

58-02 Research exposition (monographs, survey articles) pertaining to global analysis
58J40 Pseudodifferential and Fourier integral operators on manifolds
35S05 Pseudodifferential operators as generalizations of partial differential operators
35P20 Asymptotic distributions of eigenvalues in context of PDEs
58J47 Propagation of singularities; initial value problems on manifolds
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
53D50 Geometric quantization