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Quantization methods in differential equations. (English) Zbl 1079.81040
Differential and Integral Equations and Their Applications 3. London: Taylor & Francis (ISBN 0-415-27364-1). xii, 356 p. (2002).
Publisher’s description: This volume presents a systematic and mathematically rigorous exposition of methods for studying linear partial differential equations. It focuses on quantization of the corresponding objects (states, observables, and canonical transformations) in the phase space. The quantization of all three types of classical objects is carried out in a unified way with the use of a special integral transform. This book covers recent as well as established results, all treated within the framework.
Contents: Semiclassical Quantization. Quantization and Microlocalization. Quantization by the Wave Packet Transform. Maslov’s Canonical Operator and Hörmander’s Oscillatory Integrals. Topological Aspects of Quantization Conditions. The Schrödinger Equation. The Maxwell Equations. Equations with Trapping Hamiltonians. Quantization by the Method of Ordered Operators (Noncommutative Analysis). Noncommutative Analysis: Main Ideas, Definitions, and Theorems. Exactly Soluble Commutation Relations.

MSC:
58J40 Pseudodifferential and Fourier integral operators on manifolds
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
35B40 Asymptotic behavior of solutions to PDEs
35Q40 PDEs in connection with quantum mechanics
35Q60 PDEs in connection with optics and electromagnetic theory
47G30 Pseudodifferential operators
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
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