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