Methods of Noncommutative Analysis: Theory and Applications

Description

Preface I Elementary Notions of Noncommutative Analysis 1 Some Situations where Functions of Noncommuting Operators Arise 1.1 Nonautonomous Linear Differential Equations of First Order. T-Exponentials 1.2 Operators of Quantum Mechanics. Creation and Annihilation Operators 1.3 Differential and Integral Operators 1.4 Problems of Perturbation Theory 1.5 Multiplication Law in Lie Groups 1.6 Eigenfunctions and Eigenvalues of the Quantum Oscillator 1.7 T-Exponentials, Trotter Formulas, and Path Integrals 2 Functions of Noncommuting Operators: the Construction and Main Properties 2.1 Motivations 2.2 The Definition and the Uniqueness Theorem 2.3 Basic Properties 2.4 Tempered Symbols and Generators of Tempered Groups 2.5 The Influence of the Symbol Classes on the Properties of Generators 2.6 Weyl Quantization 3 Noncommutative Differential Calculus 3.1 The Derivation Formula 3.2 The Daletskii-Krein Formula 3.3 Higher-Order Expansions 3.4 Permutation of Feynman Indices 3.5 The Composite Function Formula 4 The Campbell-Hausdorff Theorem and Dynkin’s Formula 4.1 Statement of the Problem 4.2 The Commutation Operation 4.3 A Closed Formula for In (eBeA) 4.4 A Closed Formula for the Logarithm of a T-Exponential 5 Summary: Rules of “Operator Arithmetic” and Some Standard Techniques 5.1 Notation 5.2 Rules 5.3 Standard Techniques II Method of Ordered Representation 1 Ordered Representation: Definition and Main Property 1.1 Wick Normal Form 1.2 Ordered Representation and Theorem on Products 1.3 Reduction to Normal Form 2 Some Examples 2.1 Functions of the Operators x and – ihә/dә 2.2 Perturbed Heisenberg Relations 2.3 Examples of Nonlinear Commutation Relations 2.4 Lie Commutation Relations 2.5 Graded Lie Algebras 3 Evaluation of the Ordered Representation Operators 3.1 Equations for the Ordered Representation Operators 3.2 How to Obtain the Solution 3.3 Semilinear Commutation Relations 4 The Jacobi Condition and Poincaré-Birkhoff-Witt Theorem 4.1 Ordered Representation of Relation Systems and the Jacobi Condition 4.2 The Poincaré-Birkhoff-Witt Theorem 4.3 Verification of the Jacobi Condition: Two Examples 5 The Ordered Representations, Jacobi Condition, and the Yang-Baxter Equation 6 Representations of Lie Groups and Functions of Their Generators 6.1 Conditions on the Representation 6.2 Hilbert Scales 6.3 Symbol Spaces 6.4 Symbol Classes: More Suitable for Asymptotic Problems III Noncommutative Analysis and Differential Equations 1 Preliminaries 1.1 Heaviside’s Operator Method for Differential Equations with Constant Coefficients 1.2 Nonstandard Characteristics and Asymptotic Expansions 1.3 Asymptotic Expansions: Smoothness vs Parameter 1.4 Asymptotic Expansions with Respect to an Ordered Tuple of Operators 1.5 Reduction to Pseudodifferential Equations 1.6 Commutation of an h-1-Pseudodifferential Operator with an Exponential 1.7 Summary: the General Scheme 2 Difference and Difference-Differential Equations 2.1 Difference Approximations as Pseudodifferential Equations 2.2 Difference Approximations as Functions of x and δx± 2.3 Another Approach to Difference Approximations 3 Propagation of Electromagnetic Waves in Plasma 3.1 Statement of the Problem 3.2 The Construction of the Asymptotic Expansion 3.3 Analysis of the Asymptotic Solution 4 Equations with Symbols Growing at Infinity 4.1 Statement of the Problem and its Operator Interpretation 4.2 Asymptotic Solution of the Symbolic Equation 4.3 Equations with Fractional Powers of x in the Coefficients 5 Geostrophic Wind Equations 6 Degenerate Equations 6.1 Statement of the Problem 6.2 Localization of the Right-Hand Side 6.3 Solving the Equation with Localized Right-Hand Side 6.4 The Asymptotic Solution in the General Case 7 Microlocal Asymptotic Solutions for an Operator with Double Characteristics IV Functional-Analytic Background of Noncommutative Analysis 1 Topics on Convergence 1.1 What Is Actually Needed? 1.2 Polynormed Spaces and Algebras 1.3 Tensor Products 2 Symbol Spaces and Generators 2.1 Definitions 2.2 S∞ Is a Proper Symbol Space 2.3 S∞-Generators 3 Functions of Operators in Scales of Spaces 3.1 Banach Scales 3.2 S∞-Generators in Banach Scales 3.3 Functions of Feynman-Ordered Selfadjoint Operators Appendix A. Representation of Lie Algebras and Lie Groups 1 Lie Algebras and Their Representations 1.1 Lie Algebras, Bases, Structure Constants, Subalgebras 1.2 Examples of Lie Algebras 1.3 Homomorphisms, Ideals, Quotient Algebras 1.4 Representations 1.5 The Associated Representation ad. The Center of a Lie Algebra 1.6 The Ado Theorem 1.7 Nilpotent Lie Algebras 2 Lie Groups and Their Representations 2.1 Lie Groups, Subgroups, the Gleason-Montgomery-Zippin Theorem 2.2 Examples of Lie Groups 2.3 Local Lie Groups 2.4 Homomorphisms of Lie Groups, Normal Subgroups, Quotient Groups 3 Left and Right Translations. The Haar Measure 3.1 Left and Right Regular Representations 3.2 Representations of Lie Groups 4 The Relationship between Lie Groups and Lie Algebras 4.1 The Lie Algebra of a Lie Group 4.2 Examples 4.3 The Exponential Mapping, One-Parameter Subgroups, Coordinates of I and II Genera 4.4 Evaluating the Commutator with the Help of the Mapping exp 4.5 Derived Homomorphisms 4.6 Derived Representation 4.7 The Lie Group Corresponding to a Lie Algebra 4.8 The Krein-Shikhvatov Theorem Appendix B. Pseudodifferential Operators 1 Elementary Introduction 2 Symbol Spaces and Generators 3 Pseudodifferential Operators Glossary Bibliographical Remarks Bibliography Index