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This volume is concerned with a detailed description of the canonical operator method - one of the asymptotic methods of linear mathematical physics. The book is, in fact, an extension and continuation of the authors' works [59], [60], [65]. The basic ideas are summarized in the Introduction. The book consists of two parts. In the first, the theory of the canonical operator is develop ed, whereas, in the second, many applications of the canonical operator method to concrete problems of mathematical physics are presented. The authors are pleased to express their deep gratitude to S. M. Tsidilin for his valuable comments. THE AUTHORS IX INTRODUCTION 1. Various problems of mathematical and theoretical physics involve partial differential equations with a small parameter at the highest derivative terms. For constructing approximate solutions of these equations, asymptotic methods have long been used. In recent decades there has been a renaissance period of the asymptotic methods of linear mathematical physics. The range of their applicability has expanded: the asymptotic methods have been not only continuously used in traditional branches of mathematical physics but also have had an essential impact on the development of the general theory of partial differential equations. It appeared recently that there is a unified approach to a number of problems which, at first sight, looked rather unrelated.
List of contents
I Quantization of Velocity Field (the Canonical Operator).- 1. The method of Stationary phase. The Legendre Transformation.- 2. Pseudodifferential Operators.- 3. The Hamilton-Jacobi Equation. The Hamilton System.- 4. The Lagrangian Manifolds and Canonical Transformations.- 5. Fourier Transformation of a ?-Pseudo-differential Operator (the Transition to p-Representation).- 6. The Precanonical Operator (Quantization of the Velocity Field in the Small).- 7. The Index of a Curve on a Lagrangian Manifold.- 8. The Canonical Operator (Global Quantization of the Velocity Field).- 9. Global Quantization of the Velocity Field. Higher Approximations.- II Semi-Classical Approximation for Non-Relativistic and Relativistic Quantum Mechanical Equations.- 10. The Cauchy Problem with Rapidly Oscillating Initial Data for Scalar Hamiltonians.- 11. Matrix Hamiltonians.- 12. The Semi-Classical Asymptotics of the Cauchy Problem for the Schrödinger Equation.- 13. The Asymptotic Series for the Eigenvalues (Bohr's Quantization Rule).- 14. Semi-Classical Approximations for the Relativistic Dirac Equation.- References.- Index of Assumptions, Theorems, Etc..
