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, treated within the framework of a universal approach. It also includes applications and provides a useful reference text for graduate and research-level readers.
Semiclassical Quantization. Quantization and Microlocalization. Quantization by the Wave Packet Transform. Maslov's Canonical Operator and Hormander's Oscillatory Integrals. Topological Aspects of Quantization Conditions. The Schrodinger 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. Operator Algebras on Singular Manifolds. The High-Frequency Asymptotics in the Problem of Wave Propagation in Plasma. Appendices.