Operational Methods

Front Cover
Title Page
CONTENTS
PREFACE
INTRODUCTION TO OPERATIONAL CALCULUS
Sec. 1. Solution of Ordinary Differential Equations by the Heaviside Operational Method
Sec. 2. Difference Equations
Sec. 3. Solution of Systems of Differential Equations by the Heaviside Operational Method
Sec. 4. Algebra of Convergent Power Series of Noncommutative Operators
Sec. 5. Spectrum of a Pair of Ordered Operators
Sec. 6. Algebras with mu-Structures
Sec. 7. An Example of a Solution of a Differential Equation
Sec. 8. Passage of the Equation of Oscillations of a Crystal Lattice into a Wave Equation
Sec. 9. The Concept of a Quasi-Inverse Operator and Formulation of the Main Theorem
I. FUNCTIONS OF A REGULAR OPERATOR
Sec. 1. Certain Spaces of Continuous Functions and Related Spaces
Sec. 2. Embedding Theorems
Sec. 3. The Algebra of Functions of a Generator
Sec. 4. The Extension of the Class of Possible Symbols
Sec. 5. Homomorphism of Asymptotic Formulas. The Method of Stationary Phase
Sec. 6. The Spectrum of a Generator
Sec. 7. Regular Operators
Sec. 8. The Generalized Eigenfunctions and Associated Functions
Sec. 9. Self-Adjoint Operators as Transformers in the Schmidt Space*
II. CALCULUS OF NONCOMMUTATIVE OPERATORS
Sec. 1. Preliminary Definitions
Sec. 2. The Functions of Two Noncommutative Self-Adjoint Operators
Sec. 3. The Functions of Noncommutative Operators
Sec. 4. The Spectrum of a Vector-Operator
Sec. 5. Theorem on Homomorphism
Sec. 6. Problems
Sec. 7. Differentiation of the Functions of an Operator Depending ona Parameter
Sec. 8. Formulas of Commutation
Sec. 9. Growing Symbols
Sec. 10. The Factor-Spectrum
Sec. 11. The Functions of Components of a Lie Nilpotent Algebraand Their Representations
III. ASYMPTOTIC METHODS
Sec. 1. Canonical Transformations of PseudoditlerentialOperators
Sec. 2. The Homomorphism of Asymptotic Formulas
Sec. 3. The Geometrical Interpretation of the Method of Stationary Phase
Sec. 4. The Canonical Operator on an Unclosed Curve
Sec. 5. The Method of Stationary Phase
Sec. 6. The Canonical Operator on the Unclosed Curve Depending onParameters Defined Correct to 0 ( 1/ \omega )
Sec. 7. V -Objects on the Curve
Sec. 8. The Canonical Operator on the Family of Unclosed Curves
Sec. 9. The Canonical Operator on the Familyof Closed Curves
Sec. 10. An Example of Commutation of a Canonical Operator with a Hamiltonian
Sec. 11. Commutation of a Hamiltonian with a Canonical Operator
Sec. 12. The General Canonical Transformation of the Pseudodifferential Operator
IV. GENERALIZED HAMILTON-JACOBI EQUATIONS
Sec. 1. Hamilton-Jacobi Equations with Dissipation
Sec. 2. The Lagrangean Manifold with a Complex Germ
Sec. 3. \gamma -Atlases and the Dissipativity Inequality
Sec. 4. Solution of the Hamilton-Jacobi Equation with Dissipation
Sec. 5. Preservation of the Dissipativity Inequality. Bypassing Focuses Operation.
Sec. 6. Solution of Transfer Equation with Dissipation
V. CANONICAL OPERATOR ON A LAGRANGEAN MANIFOLD WITH A COMPLEX GERM AND PROOF OF THE MAIN THEOREM
Sec. 1. Quantum Bypassing Focuses Operation
Sec. 2. Commutation Formulas for a Complex Exponential and a Hamiltonian
Sec. 3. C -Lagrangean Manifolds and the Index of a Complex Germ
Sec. 4. Canonical Operator
Sec. 5. Proof of the Main Theorem
Appendix to Sec. 5
Sec. 6. Cauchy Problem for Systems with Complex Characteristics
Sec. 7. Quasi-Inverse of Operators with Matrix Symbol
APPENDIX Spectral Expansion of T-products
Index