Methods of Noncommutative Analysis

, by Vladimir E. Nazaikinskii; Victor E. Shatalov; Boris Yu. Sternin

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ISBN: 9783110146325 | 3110146320
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Copyright: 6/24/2011

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Supplemental Materials

Preface

Elementary Notions of Noncommutative Analysis

(93)

Some Situations where Functions of Noncommuting Operators Arise

(22)

Nonautonomous Linear Differential Equations of First Order. T-Exponentials

(3)

Operators of Quantum Mechanics. Creation and Annihilation Operators

(3)

Differential and Integral Operators

(3)

Problems of Perturbation Theory

(4)

Multiplication Law in Lie Groups

(2)

Eigenfunctions and Eigenvalues of the Quantum Oscillator

(4)

T-Exponentials, Trotter Formulas, and Path Integrals

(3)

Functions of Noncommuting Operators: the Construction and Main Properties

(28)

Motivations

(3)

The Definition and the Uniqueness Theorem

(7)

Basic Properties

(9)

Tempered Symbols and Generators of Tempered Groups

(3)

The Influence of the Symbol Classes on the Properties of Generators

(3)

Weyl Quantization

(3)

Noncommutative Differential Calculus

(19)

The Derivation Formula

(2)

The Daletskii-Krein Formula

(1)

Higher-Order Expansions

(5)

Permutation of Feynman Indices

(6)

The Composite Function Formula

(4)

The Campbell-Hausdorff Theorem and Dynkin's Formula

(14)

Statement of the Problem

(1)

The Commutation Operation

(3)

A Closed Formula for ln (eB eA)

(3)

A Closed Formula for the Logarithm of a T-Exponential

(7)

Summary: Rules of ``Operator Arithmetic'' and Some Standard Techniques

(10)

Notation

(1)

Rules

(1)

Standard Techniques

(7)

Method of Ordered Representation

(77)

Ordered Representation: Definition and Main Property

(10)

Wick Normal Form

(3)

Ordered Representation and Theorem on Products

(2)

Reduction to Normal Form

(5)

Some Examples

104

(13)

Functions of the Operators x and -ih∂/∂x

105

(2)

Perturbed Heisenberg Relations

107

(1)

Examples of Nonlinear Commutation Relations

108

(2)

Lie Commutation Relations

110

(5)

Graded Lie Algebras

115

(2)

Evaluation of the Ordered Representation Operators

117

(15)

Equations for the Ordered Representation Operators

117

(4)

How to Obtain the Solution

121

(4)

Semilinear Commutation Relations

125

(7)

The Jacobi Condition and Poincare-Birkhoff-Witt Theorem

132

(12)

Ordered Representation of Relation Systems and the Jacobi Condition

133

(5)

The Poincare-Birkhoff-Witt Theorem

138

(4)

Verification of the Jacobi Condition: Two Examples

142

(2)

The Ordered Representations, Jacobi Condition, and the Yang-Baxter Equation

144

(12)

Representations of Lie Groups and Functions of Their Generators

156

(15)

Conditions on the Representation

156

(2)

Hilbert Scales

158

(3)

Symbol Spaces

161

(6)

Symbol Classes: More Suitable for Asymptotic Problems

167

(4)

Noncommutative Analysis and Differential Equations

171

(71)

Preliminaries

171

(22)

Heaviside's Operator Method for Differential Equations with Constant Coefficients

174

(5)

Nonstandard Characteristics and Asymptotic Expansions

179

(3)

Asymptotic Expansions: Smoothness vs Parameter

182

(3)

Asymptotic Expansions with Respect to an Ordered Tuple of Operators

185

(1)

Reduction to Pseudodifferential Equations

186

(3)

Commutation of an h-1-Pseudodifferential Operator with an Exponential

189

(2)

Summary: the General Scheme

191

(2)

Difference and Difference-Differential Equations

193

(7)

Difference Approximations as Pseudodifferential Equations

194

(2)

Difference Approximations as Functions of x and δ±x

196

(2)

Another Approach to Difference Approximations

198

(2)

Propagation of Electromagnetic Waves in Plasma

200

(8)

Statement of the Problem

200

(2)

The Construction of the Asymptotic Expansion

202

(3)

Analysis of the Asymptotic Solution

205

(3)

Equations with Symbols Growing at Infinity

208

(8)

Statement of the Problem and its Operator Interpretation

208

(2)

Asymptotic Solution of the Symbolic Equation

210

(2)

Equations with Fractional Powers of x in the Coefficients

212

(4)

Geostrophic Wind Equations

216

(8)

Degenerate Equations

224

(9)

Statement of the Problem

224

(1)

Localization of the Right-Hand Side

225

(4)

Solving the Equation with Localized Right-Hand Side

229

(3)

The Asymptotic Solution in the General Case

232

(1)

Microlocal Asymptotic Solutions for an Operator with Double Characteristics

233

(9)

Functional-Analytic Background of Noncommutative Analysis

242

(45)

Topics on Convergence

242

(18)

What Is Actually Needed?

242

(7)

Polynormed Spaces and Algebras

249

(8)

Tensor Products

257

(3)

Symbol Spaces and Generators

260

(10)

Definitions

260

(3)

S∞ Is a Proper Symbol Space

263

(5)

S∞-Generators

268

(2)

Functions of Operators in Scales of Spaces

270

(17)

Banach Scales

270

(3)

S∞-Generators in Banach Scales

273

(5)

Functions of Feynman-Ordered Selfadjoint Operators

278

(9)

Appendix A. Representation of Lie Algebras and Lie Groups

287

(24)

1 Lie Algebras and Their Representations

287

(5)

1.1 Lie Algebras, Bases, Structure Constants, Subalgebras

287

(1)

1.2 Examples of Lie Algebras

288

(1)

1.3 Homomorphisms, Ideals, Quotient Algebras

289

(1)

1.4 Representations

290

(1)

1.5 The Associated Representation ad. The Center of a Lie Algebra

291

(1)

1.6 The Ado Theorem

291

(1)

1.7 Nilpotent Lie Algebras

292

(1)

2 Lie Groups and Their Representations

292

(3)

2.2 Lie Groups, Subgroups, the Gleason-Montgomery-Zippin Theorem

292

(1)

2.3 Examples of Lie Groups

293

(1)

2.4 Local Lie Groups

294

(1)

2.5 Homomorphisms of Lie Groups, Normal Subgroups, Quotient Groups

294

(1)

3 Left and Right Translations. The Haar Measure

295

(2)

3.1 Left and Right Regular Representations

295

(1)

3.2 Representations of Lie Groups

296

(1)

4 The Relationship between Lie Groups and Lie Algebras

297

(14)

4.1 The Lie Algebra of a Lie Group

297

(1)

4.2 Examples

298

(1)

4.3 The Exponential Mapping, One-Parameter Subgroups, Coordinates of I and II Genera

299

(2)

4.4 Evaluating the Commutator with the Help of the Mapping exp

301

(1)

4.5 Derived Homomorphisms

302

(1)

4.6 Derived Representation

303

(2)

4.7 The Lie Group Corresponding to a Lie Algebra

305

(2)

4.8 The Krein-Shikhvatov Theorem

307

(4)

Appendix B. Pseudodifferential Operators

311

(16)

1 Elementary Introduction

311

(6)

2 Symbol Spaces and Generators

317

(4)

3 Pseudodifferential Operators

321

(6)

Glossary

327

(24)

Bibliographical Remarks

351

(6)

Bibliography

357

(14)

Index

371

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