Title Methods of noncommutative analysis : theory and applications / Vladimir E. Nazaikinskii, Victor E. Shatalov, Boris Yu. Sternin

Published Berlin ; New York : Walter de Gruyter, 1995

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Description 1 online resource (x, 373 pages) Series De Gruyter studies in mathematics ; 22 De Gruyter studies in mathematics ; 22. 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 Bibliography Includes bibliographical references and index Notes English Subject Geometry, Differential. Noncommutative algebras. Mathematical physics. MATHEMATICS -- Functional Analysis. Geometry, Differential Mathematical physics Noncommutative algebras Form Electronic book Author Shatalov, V. E. (Viktor Evgenʹevich) Sternin, B. I︠U︡ ISBN 9783110813548 3110813548 1306275261 9781306275262

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