Methods of noncommutative analysis : theory and applications /

Methods of noncommutative analysis : theory and applications /

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Bibliographic Details
Main Author: Nazaĭkinskiĭ, V. E.
Other Authors: Shatalov, V. E., Sternin, B. I︠U︡
Format: Electronic eBook
Language:English
Published: Berlin ; New York : Walter de Gruyter, 1995.
Series:De Gruyter studies in mathematics ; 22.
Subjects:
Geometry, Differential.
Noncommutative algebras.
Mathematical physics.
Online Access:CONNECT
Table of Contents:
  • 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 Properties2.1 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 Dynkinâ€?s Formula4.1 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 Arithmeticâ€? 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 Examples2.1 Functions of the Operators x and â€? ihÓ?/dÓ?
  • 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 Theorem4.3 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 Heavisideâ€?s Operator Method for Differential Equations with Constant Coefficients

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