Methods of noncommutative analysis: theory and applications
Gespeichert in:
| Beteiligte Personen: | , , |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Berlin [u.a.]
de Gruyter
1996
|
| Schriftenreihe: | De Gruyter studies in mathematics
22 |
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006908590&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Beschreibung: | Literaturverz. S. 357 - 369 |
| Umfang: | X, 373 S. |
| ISBN: | 3110146320 |
Internformat
MARC
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| 100 | 1 | |a Nazajkinskij, Vladimir E. |d 1955- |e Verfasser |0 (DE-588)121444643 |4 aut | |
| 245 | 1 | 0 | |a Methods of noncommutative analysis |b theory and applications |c Vladimir E. Nazaikinskii ; Victor E. Shatalov ; Boris Yu. Sternin |
| 264 | 1 | |a Berlin [u.a.] |b de Gruyter |c 1996 | |
| 300 | |a X, 373 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a De Gruyter studies in mathematics |v 22 | |
| 500 | |a Literaturverz. S. 357 - 369 | ||
| 650 | 4 | |a Differentialgleichung - Nicht vertauschbarer Operator | |
| 650 | 4 | |a Funktionalanalysis - Nicht vertauschbarer Operator | |
| 650 | 4 | |a Nichtkommutative mikrolokale Analysis | |
| 650 | 0 | 7 | |a Nicht vertauschbarer Operator |0 (DE-588)4303614-4 |2 gnd |9 rswk-swf |
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| 700 | 1 | |a Šatalov, Viktor E. |d 1945- |e Verfasser |0 (DE-588)120484544 |4 aut | |
| 700 | 1 | |a Sternin, Boris Ju. |d 1939- |e Verfasser |0 (DE-588)120484536 |4 aut | |
| 830 | 0 | |a De Gruyter studies in mathematics |v 22 |w (DE-604)BV000005407 |9 22 | |
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Datensatz im Suchindex
| DE-BY-TUM_call_number | 0001 04.2001 A 7147 0001 96 A 2825 |
|---|---|
| DE-BY-TUM_katkey | 777136 |
| DE-BY-TUM_location | Mag |
| DE-BY-TUM_media_number | 040004441792 040001900747 |
| _version_ | 1821938792190181376 |
| adam_text | Contents
Preface v
I Elementary Notions of Noncommutative Analysis 1
1 Some Situations where Functions of Noncommuting Operators Arise . 1
1.1 Nonautonomous Linear Differential Equations of First Order.
T Exponentials 1
1.2 Operators of Quantum Mechanics. Creation and Annihilation
Operators 4
1.3 Differential and Integral Operators 7
1.4 Problems of Perturbation Theory 10
1.5 Multiplication Law in Lie Groups 14
1.6 Eigenfunctions and Eigenvalues of the Quantum Oscillator . . 16
1.7 r Exponentials, Trotter Formulas, and Path Integrals 20
2 Functions of Noncommuting Operators: the Construction and Main
Properties 23
2.1 Motivations 23
2.2 The Definition and the Uniqueness Theorem 26
2.3 Basic Properties 33
2.4 Tempered Symbols and Generators of Tempered Groups ... 42
2.5 The Influence of the Symbol Classes on the Properties of Gen¬
erators 45
2.6 Weyl Quantization 48
3 Noncommutative Differential Calculus 51
3.1 The Derivation Formula 52
3.2 The Daletskii Krein Formula 54
3.3 Higher Order Expansions 55
3.4 Permutation of Feynman Indices 60
3.5 The Composite Function Formula 66
4 The Campbell Hausdorff Theorem and Dynkin s Formula 70
4.1 Statement of the Problem 70
4.2 The Commutation Operation 71
4.3 A Closed Formula for In (eBeA) 74
4.4 A Closed Formula for the Logarithm of a T Exponential ... 77
5 Summary: Rules of Operator Arithmetic and Some Standard Tech¬
niques 84
viii CONTENTS
5.1 Notation 85
5.2 Rules 86
5.3 Standard Techniques 87
II Method of Ordered Representation 94
1 Ordered Representation: Definition and Main Property 94
1.1 Wick Normal Form 94
1.2 Ordered Representation and Theorem on Products 97
1.3 Reduction to Normal Form 99
2 Some Examples 104
2.1 Functions of the Operators x and —ihd/dx 105
2.2 Perturbed Heisenberg Relations 107
2.3 Examples of Nonlinear Commutation Relations 108
2.4 Lie Commutation Relations 110
2.5 Graded Lie Algebras 115
3 Evaluation of the Ordered Representation Operators 117
3.1 Equations for the Ordered Representation Operators 117
3.2 How to Obtain the Solution 121
3.3 Semilinear Commutation Relations 125
4 The Jacobi Condition and Poincare Birkhoff Witt Theorem 132
4.1 Ordered Representation of Relation Systems and the Jacobi
Condition 133
4.2 The Poincare Birkhoff Witt Theorem 138
4.3 Verification of the Jacobi Condition: Two Examples 142
5 The Ordered Representations, Jacobi Condition, and the Yang Baxter
Equation 144
6 Representations of Lie Groups and Functions of Their Generators . . .156
6.1 Conditions on the Representation 156
6.2 Hilbert Scales 158
6.3 Symbol Spaces 161
6.4 Symbol Classes: More Suitable for Asymptotic Problems . . .167
III Noncommutative Analysis and Differential Equations 171
1 Preliminaries 171
1.1 Heaviside s Operator Method for Differential Equations with
Constant Coefficients 174
1.2 Nonstandard Characteristics and Asymptotic Expansions . . . 179
1.3 Asymptotic Expansions: Smoothness vs Parameter 182
1.4 Asymptotic Expansions with Respect to an Ordered Tuple of
Operators 185
1.5 Reduction to Pseudodifferential Equations 186
1.6 Commutation of an A 1 Pseudodifferential Operator with an
Exponential 189
CONTENTS ix
1.7 Summary: the General Scheme 191
2 Difference and Difference Differential Equations 193
2.1 Difference Approximations as Pseudodifferential Equations . .194
2.2 Difference Approximations as Functions of x and Sf 196
2.3 Another Approach to Difference Approximations 198
3 Propagation of Electromagnetic Waves in Plasma 200
3.1 Statement of the Problem 200
3.2 The Construction of the Asymptotic Expansion 202
3.3 Analysis of the Asymptotic Solution 205
4 Equations with Symbols Growing at Infinity 208
4.1 Statement of the Problem and its Operator Interpretation . . . 208
4.2 Asymptotic Solution of the Symbolic Equation 210
4.3 Equations with Fractional Powersoft in the Coefficients . . .212
5 Geostrophic Wind Equations 216
6 Degenerate Equations 224
6.1 Statement of the Problem 224
6.2 Localization of the Right Hand Side 225
6.3 Solving the Equation with Localized Right Hand Side 229
6.4 The Asymptotic Solution in the General Case 232
7 Microlocal Asymptotic Solutions for an Operator with Double Char¬
acteristics 233
IV Functional Analytic Background of Noncommutative Analysis 242
1 Topics on Convergence 242
1.1 What Is Actually Needed? 242
1.2 Polynormed Spaces and Algebras 249
1.3 Tensor Products 257
2 Symbol Spaces and Generators 260
2.1 Definitions 260
2.2 5°° Is a Proper Symbol Space 263
2.3 ^ Generators 268
3 Functions of Operators in Scales of Spaces 270
3.1 Banach Scales 270
3.2 S°° Generators in Banach Scales 273
3.3 Functions of Feynman OrderedSelfadjoint Operators 278
Appendix A. Representation of Lie Algebras and Lie Groups 287
1 Lie Algebras and Their Representations 287
1.1 Lie Algebras, Bases, Structure Constants, Subalgebras . . . .287
1.2 Examples of Lie Algebras 288
1.3 Homomorphisms, Ideals, Quotient Algebras 289
1.4 Representations 290
1.5 The Associated Representation ad. The Center of a Lie Algebra 291
x CONTENTS
1.6 The Ado Theorem 291
1.7 Nilpotent Lie Algebras 292
2 Lie Groups and Their Representations 292
2.1 Lie Groups, Subgroups, the Gleason Montgomery Zippin The¬
orem 292
2.2 Examples of Lie Groups 293
2.3 Local Lie Groups 294
2.4 Homomorphisms of Lie Groups, Normal Subgroups, Quotient
Groups 294
3 Left and Right Translations. The Haar Measure 295
3.1 Left and Right Regular Representations 295
3.2 Representations of Lie Groups 296
4 The Relationship between Lie Groups and Lie Algebras 297
4.1 The Lie Algebra of a Lie Group 297
4.2 Examples 298
4.3 The Exponential Mapping, One Parameter Subgroups, Coor¬
dinates of I and II Genera 299
4.4 Evaluating the Commutator with the Help of the Mapping exp 301
4.5 Derived Homomorphisms 302
4.6 Derived Representation 303
4.7 The Lie Group Corresponding to a Lie Algebra 305
4.8 The Krein Shikhvatov Theorem 307
Appendix B. Pseudodifferential Operators 311
1 Elementary Introduction 311
2 Symbol Spaces and Generators 317
3 Pseudodifferential Operators 321
Glossary 327
Bibliographical Remarks 351
Bibliography 357
Index 371
|
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| ctrlnum | (OCoLC)246743379 (DE-599)BVBBV010376855 |
| discipline | Mathematik |
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| id | DE-604.BV010376855 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-20T09:52:59Z |
| institution | BVB |
| isbn | 3110146320 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-006908590 |
| oclc_num | 246743379 |
| open_access_boolean | |
| owner | DE-20 DE-739 DE-12 DE-824 DE-91 DE-BY-TUM DE-634 DE-11 DE-188 |
| owner_facet | DE-20 DE-739 DE-12 DE-824 DE-91 DE-BY-TUM DE-634 DE-11 DE-188 |
| physical | X, 373 S. |
| publishDate | 1996 |
| publishDateSearch | 1996 |
| publishDateSort | 1996 |
| publisher | de Gruyter |
| record_format | marc |
| series | De Gruyter studies in mathematics |
| series2 | De Gruyter studies in mathematics |
| spellingShingle | Nazajkinskij, Vladimir E. 1955- Šatalov, Viktor E. 1945- Sternin, Boris Ju. 1939- Methods of noncommutative analysis theory and applications De Gruyter studies in mathematics Differentialgleichung - Nicht vertauschbarer Operator Funktionalanalysis - Nicht vertauschbarer Operator Nichtkommutative mikrolokale Analysis Nicht vertauschbarer Operator (DE-588)4303614-4 gnd |
| subject_GND | (DE-588)4303614-4 |
| title | Methods of noncommutative analysis theory and applications |
| title_auth | Methods of noncommutative analysis theory and applications |
| title_exact_search | Methods of noncommutative analysis theory and applications |
| title_full | Methods of noncommutative analysis theory and applications Vladimir E. Nazaikinskii ; Victor E. Shatalov ; Boris Yu. Sternin |
| title_fullStr | Methods of noncommutative analysis theory and applications Vladimir E. Nazaikinskii ; Victor E. Shatalov ; Boris Yu. Sternin |
| title_full_unstemmed | Methods of noncommutative analysis theory and applications Vladimir E. Nazaikinskii ; Victor E. Shatalov ; Boris Yu. Sternin |
| title_short | Methods of noncommutative analysis |
| title_sort | methods of noncommutative analysis theory and applications |
| title_sub | theory and applications |
| topic | Differentialgleichung - Nicht vertauschbarer Operator Funktionalanalysis - Nicht vertauschbarer Operator Nichtkommutative mikrolokale Analysis Nicht vertauschbarer Operator (DE-588)4303614-4 gnd |
| topic_facet | Differentialgleichung - Nicht vertauschbarer Operator Funktionalanalysis - Nicht vertauschbarer Operator Nichtkommutative mikrolokale Analysis Nicht vertauschbarer Operator |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006908590&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000005407 |
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| Exemplar 2 | Ausleihbar Am Standort |