Quantization Methods in the Theory of Differential Equations

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.

Theoretical physics, especially quantum mechanics, has always both supplied ideas to and used the results of the theory of differential equations. As a product of the interaction between the two disciplines, there arose a new and very productive science, known as mathematical physics. Not only has its development led to new results in the natural sciences, but it has also given an impetus for new ideas in other mathematical fields, such as representation theory, algebraic topology, and differential geometry.

The present book gives a systematic mathematically rigorous exposition of methods for studying linear partial differential equations (mostly of the wave type) on the basis of quantization of appropriate objects in phase space. Here we consider two types of quantization, namely, the semiclassical (asymptotic) quantization based on the wave packet transform and quantization of, Poisson brackets based on noncommutative analysis. The wave packet transform determines the quantization of all three types of classical objects (states, observables, and canonical transformations), which is carried out in a unified way. Quantization by means of noncommutative analysis, a unique tool permitting one to treat functions of generally noncommuting operator arguments in much the same way as usual functions, unifies the procedure of obtaining diverse asymptotic, including simultaneous asymptotic with respect to a set of parameters.

A wide range of applications is considered, ranging from problems in theory of differential equations on singular manifolds to some problems of theoretical physics, e.g., propagation of electromagnetic waves in plasma or ionosphere channelling of high-frequency radio signals. Approximate
and asymptotic solutions of three-dimensional Maxwell equations are constructed, etc.

The exposition moves gradually from the simple to the complex. Numerous examples are included to help the reader understand the material.
The book is intended for a wide readership, including undergraduates, graduate students, and scientists specializing in differential equations, applied mathematics, and mathematical and theoretical physics.