Operational methods

Page de titre......Page 1
Preface......Page 5
1. Solution of Ordinary Differential Equations by the Heaviside Operational Method......Page 11
2. Difference Equations......Page 18
3. Solution of Systems of Differential Equations by the Heaviside Operational Method......Page 20
4. Algebra of Convergent Power Series of Noncommutative Operators......Page 22
5. Spectrum of a Pair of Ordered Operators......Page 33
6. Aigebras with µ-Structures......Page 38
7. An Example of a Solution of a Differential Equation......Page 54
8. Passage of the Equation of Oscillations of a Crystal Lattice into a Wave Equation......Page 56
9. The Concept of a Quasi-Inverse Operator and Formulation of the Main Theorem......Page 98
Chapter 1 Functions of a Regular Operator......Page 145
1. Certain Spaces of Continuous Functions and Related Spaces......Page 147
2. Embedding Theorems......Page 152
3. The Algebra of Functions of a Generator......Page 156
4. The Extension of the Class of Possible Symbols......Page 171
5. Homomorphism of Asymptotic Formulas. The Method of Stationary Phase......Page 179
6. The Spectrum of a Generator......Page 186
7. Regular Operators......Page 192
8. The Generalized Eigenfunctions and Associated Functions......Page 196
9. Self-Adjoint Operators as Transformers in the Schmidt Space......Page 203
1. Preliminary Definitions......Page 208
2. The Functions of Two Noncommutative Self-Adjoint Operators......Page 222
3. The Functions of Noncommutative Operators......Page 226
4. The Spectrum of a Vector-Operator......Page 229
5. Theorem on Homomorphism......Page 237
6. Problems......Page 240
7. Differentiation of the Functions of an Operator Depending on a Parameter......Page 249
8. Formulas of Commutation......Page 254
9. Growing Symbols......Page 259
10. The Factor-Spectrum......Page 263
11. The Functions of Components of a Lie Nilpotent Algebra and Their Representations......Page 264
1. Canonical Transformations of Pseudodifferential Operators......Page 271
2.. The Homomorphism of Asymptotic Formulas......Page 292
3. The Geometrical Interpretation of the Method of Stationary Phase......Page 299
4. The Canonical Operator on an Unclosed Curve......Page 301
5. The Method of Stationary Phase......Page 310
6. The Canonical Operator on the Unclosed Curve Depending on Parameters Defined Correct to 0(1/ω)......Page 313
7. V-Objects on the Curve......Page 319
8. The Canonical Operator on the Family of Unclosed Curves......Page 325
9. The Canonical Operator on the Family of Closed Curves......Page 331
10. An Example of Commutation of a Canonical Operator with a Hamiltonian......Page 337
11. Commutation of a Hamiltonian with a Canonical Operator......Page 344
12. The General Canonical Transformation of the Pseudodifferential Operator......Page 346
Chapter IV Generalized Hamilton-Jacobi Equations......Page 353
1. Hamilton-Jacobi Equations with Dissipation......Page 354
2. The Lagrangean Manifold with a Complex Germ......Page 358
3. γ-Atlases and the Dissipativity Inequality......Page 370
4. Solution of the Hamilton-Jacobi Equation with Dissipation......Page 376
5. Preservation of the Dissipativity Inequality. Bypassing Focuses Operation......Page 384
6. Solution of Transfer Equation with Dissipation......Page 399
1. Quantum Bypassing Focuses Operation......Page 417
2. Commutation Formulas for a Complex Exponential and a Hamiltonian......Page 438
3. C-Lagrangean Manifolds and the Index of a Complex Germ......Page 450
4. Canonical Operator......Page 467
5. Proof of the Main Theorem......Page 480
Appendix to 5......Page 491
6. Cauchy Problem for Systems with Complex Characteristics......Page 501
7. Quasi-Inverse of Operators with Matrix Symbols......Page 517
Appendix. Spectral Expansion of T-products......Page 543
Index......Page 555