## Operational Methods – Maslov

In this post, we will see the book Operational Methods by V. P. Maslov. …This book is devoted to one, but sufficiently general operational method, which absorbs many operational methods known to date and allows for the uniform solution of both classical problems, involving differential equations with partial derivatives, and the absolutely new problems of mathematical physics, including those connected with non-linear equations in partial derivatives.

…This book on operational methods should be accessible to senior course students of mathematics and physics faculties at universities and departments of applied mathematics. This means that only a knowledge of classical analysis is required of the reader. The book provides explanations in sufficient volume of such concepts as the theory of Banach algebras of distributions (Chapter I), the theory of linear differential and difference equations (Sees. 1, 2, and 3 of Introduction), the theory of non-linear equations of the first order with partial derivatives (Chapter IV). This material may be also of use to the reader who is already familiar with these questions, because rather often it is not presented in traditional style, and adapted for further reference. The reader who studies the book thoroughly will be equipped to carry on independent research in the modern theory of linear, non-linear differential and differential-difference equations with partial derivatives.

…This book has been written in such a way as to serve the widest possible circle of readers. It is suitable for two methods of study. The reader, who seeks to avoid fine assessments and passing to the limit and only wishes to master the practical techniques for obtaining asymptotic solutions, may omit that part of the book which is devoted to functional analysis.

…The most effective way of mastering the subject, however, consists rather in first reading Introduction and then reading all the book in succession. The reader should nevertheless be warned that all these methods are not at all easy, because the book provides a new operational calculus-the calculus of ordered operators.

The book was translated from the Russian by V. Golo, N. Kulman and G. Voropaeva and was published by Mir in 1976.

Credits to the original uploader for the scan, in this link we have converted to pdf from djvu, added bookmarks and cover.

CONTENTS
Preface 7
Introduction to Operational Calculus 13
Sec. 1. Solution of Ordinary Differential Equations by the Heaviside Operational Method 13
Sec. 2. Difference Equations 20
Sec. 3. Solution of Systems of Differential Equations by the Heaviside Operational Method 22
Sec. 4. Algebra of Convergent Power Series of Noncommutative Operators 24
Sec. 5. Spectrum of a Pair of Ordered Operators 35
Sec. 6. Algebras with \mu-Structures 40
Sec. 7. An Example of a Solution of a Differential Equation 56
Sec. 8. Passage of the Equation of Oscillations of a Crystal Lattice into a Wave Equation 58
Sec. 9. The Concept of a Quasi-Inverse Operator and Formulation of
the Main Theorem 100

Chapter I Functions of a Regular Operator 147

Sec. 1. Certain Spaces of Continuous Functions and Related Spaces 149
Sec. 2. Embedding Theorems 154
Sec. 3. The Algebra of Functions of a Generator 158
Sec. 4. The Extension of the Class of Possible Symbols 173
Sec. 5. Homomorphism of Asymptotic Formulas. The Method of Stationary Phase 181
Sec. 6. The Spectrum of a Generator 188
Sec. 7. Regular Operators 194
Sec. 8. The Generalized Eigenfunctions and Associated Functions 198
Sec. 9. Self-Adjoint Operators as Transformers in the Schmidt Space 205

Chapter II Calculus of Noncommutative Operators 210
Sec. 1. Preliminary Definitions 210
Sec. 2. The Functions of Two Noncommutative Self-Adjoint Operators 224
Sec. 3. The Functions of Noncommutative Operators 228
Sec. 4. The Spectrum of a Vector-Operator 231
Sec. 5. Theorem on Homomorphism 239
Sec. 6. Problems 242
Sec. 7. Differentiation of the Functions of an Operator Depending on a Parameter 251
Sec. 8. Formulas of Commutation 256
Sec. 9. Growing Symbols 261
Sec. 10. The Factor-Spectrum 265
Sec. 11. The Functions of Components of a Lie Nilpotent Algebra and Their Representations 266

Chapter III Asymptotic Methods 273

Sec. 1. Canonical Transformations of Pseudodifferential Operators 273
Sec. 2. The Homomorphism of Asymptotic Formulas 294
Sec. 3. The Geometrical Interpretation of the Method of Stationary
Phase 301
Sec. 4. The Canonical Operator on an Unclosed Curve 303
Sec. 5. The Method of Stationary Phase 312
Sec. 6. The Canonical Operator on the Unclosed Curve Depending on Parameters Defined Correct to 0 ( 1/\omega ) 315
Sec. 7. V-Objects on the Curve 321
Sec. 8. The Canonical Operator on the Family of Unclosed Curves 327
Sec. 9. The Canonical Operator on the Family of Closed Curves 333
Sec. 10. An Example of Commutation of a Canonical Operator with a Hamiltonian 339
Sec. 11. Commutation of a Hamiltonian with a Canonical Operator 346
Sec. 12. The General Canonical Transformation of the Pseudodifferential Operator 348

Chapter IV Generalized Hamilton-Jacobi Equations 355

Sec. 1. Hamilton-Jacobi Equations with Dissipation 356
Sec. 2. The Lagrangean Manifold with a Complex Germ 360
Sec. 3. y-Atlases and the Dissipativity Inequality 372
Sec. 4. Solution of the Hamilton-Jacobi Equation with Dissipation 378
Sec. 5. Preservation of the Dissipativity Inequality. Bypassing Focuses Operation 386
Sec. 6. Solution of Transfer Equation with Dissipation 401

Chapter V Canonical Operator on a Lagrangean Manifold with a Complex Germ and Proof of the Main Theorem 419

Sec. 1. Quantum Bypassing Focuses Operation 419
Sec. 2. Commutation Formulas for a Complex Exponential and a Hamiltonian 440
Sec. 3. C-Lagrangean Manifolds and the Index of a Complex Germ 452
Sec. 4. Canonical Operator 469
Sec. 5. Proof of the Main Theorem 482
Appendix to Sec. 5 493
Sec. 6. Cauchy Problem for Systems with Complex Characteristics 503
Sec. 7. Quasi-Inverse of Operators with Matrix Symbols 519
Appendix. Spectral Expansion of T-products 545
Index 557 