Differential equations on singular manifolds: semiclassical theory and operator algebras
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | German |
| Published: |
Berlin [u.a.]
Wiley-VCH
1998
|
| Edition: | 1. ed. |
| Series: | Mathematical topics
15 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | 376 S. graph. Darst. |
| ISBN: | 3527400869 |
Staff View
MARC
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| 245 | 1 | 0 | |a Differential equations on singular manifolds |b semiclassical theory and operator algebras |c Bert-Wolfgang Schulze ; Boris Sternin ; Victor Shatalov |
| 250 | |a 1. ed. | ||
| 264 | 1 | |a Berlin [u.a.] |b Wiley-VCH |c 1998 | |
| 300 | |a 376 S. |b graph. Darst. | ||
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Record in the Search Index
| _version_ | 1804126857237364736 |
|---|---|
| adam_text | Contents
0 Introduction. Examples and Motivation 13
0.1 What Is a Manifold with Singularities? 13
0.1.1 A circular cone 15
0.1.2 A circular cusp 16
0.1.3 Conclusions 18
0.2 The Semiclassical Approximation 19
0.2.1 The conical case 19
0.2.2 The Stokes phenomenon. Generalizations 28
0.2.3 The cuspidal case 30
0.3 Finiteness Theorems (the Fredholm Property) 38
0.3.1 Asymptotic behavior and statement of the problem 38
0.3.2 Operator algebras 41
0.4 Conclusions 47
1 Generalities 51
1 Structure Rings on Singular Manifolds 53
1.1 General Considerations 53
1.1.1 Local rings 53
1.1.2 Differential operators and Riemannian metrics 55
1.2 Power Stabilization (Cuspidal Points) 58
1.2.1 The local ring 58
1.2.2 Differential operators and Riemannian metrics 58
1.3 Exponential Stabilization (Conical Points) 60
1.3.1 The local ring 60
1.3.2 Differential operators and Riemannian metrics 61
1.4 Exponential Stabilization of Arbitrary Degree 62
1.4.1 The local ring 62
1.4.2 Differential operators and Riemannian metrics 63
1.5 Strong Exponential Stabilization 64
1.5.1 The local ring 64
8 Contents
1.5.2 Differential operators and Riemannian metrics 65
1.6 General Types of Singularities 66
1.6.1 Conification 66
1.6.2 Edgification 67
2 Interaction of Asymptotic Expansions 69
2.1 Examples 70
2.2 The General Statement 73
2.3 The Two Dimensional Case 75
2.3.1 Propagation of singularities 75
2.3.2 Asymptotics near the intersection 77
2.4 The Multidimensional Case 81
2.4.1 Propagation of singularities 81
2.4.2 Asymptotics near the intersection 82
3 Resurgent Analysis of Functions of Polynomial Growth 85
3.1 The Resurgent Representation 85
3.1.1 Definition and main properties 85
3.1.2 Invertibility properties of the resurgent representation 91
3.2 Asymptotic Expansions and the Stokes Phenomenon 92
3.2.1 Resurgent functions with simple singularities 93
3.2.2 The Stokes phenomenon and the connection homomorphism . . 95
3.2.3 Conditions of single valuedness 99
3.2.4 Asymptotic expansions near focal points 106
3.3 A Classification of Asymptotic Expansions of Functions of Polynomial
Growth 110
3.3.1 The generalized resurgent representation Ill
3.3.2 Types of asymptotic expansions 113
II Elliptic Equations 121
4 Asymptotic Solutions on Manifolds with Conical Singularities 123
4.1 The Construction of Resurgent Solutions 124
4.1.1 Statement of the problem 124
4.1.2 Reduction to a resurgent equation 125
4.1.3 Solving the resurgent equations 126
4.2 Applications and Examples 129
4.2.1 Applications 129
4.2.2 Elliptic equations on the cone 130
4.2.3 Elliptic equations on a manifold with an edge 133
Contents 9
5 Asymptotic Solutions on Manifolds with Cusp Type Singularities 135
5.1 Examples 135
5.1.1 The cusp of order 1 137
5.1.2 The cusp of order 2 140
5.2 Formal Theory 143
5.2.1 The general asymptotic expansion 144
5.2.2 Analysis of the asymptotic expansion 149
5.2.3 Explicit computation of the coefficients 151
5.3 The Construction of Resurgent Solutions 159
5.3.1 The case of a simple cusp 160
5.3.2 The case of a cusp of higher multiplicity 165
6 Asymptotic Solutions on Manifolds with Corner Type Singularities 171
6.1 Example 171
6.2 The Construction of Resurgent Solutions 178
6.2.1 Statement of the problem 178
6.2.2 The solvability theorem 182
6.2.3 Investigation of the singularity set of the solution 184
6.2.4 Asymptotics of solutions near the vertex 186
6.2.5 The case of resurgent functions with simple singularities .... 191
6.3 Two Dimensional Problems 194
6.3.1 Resurgent solutions 194
6.3.2 The solvability of an analytic family of one dimensional
problems 196
7 General Asymptotic Theory 201
7.1 Resurgent Analysis 202
7.1.1 Preliminaries 202
7.1.2 Analytic groups and integral representations 204
7.1.3 Resurgent elements of the algebra 211
7.1.4 Examples 215
7.1.5 The parametric case and the Stokes phenomenon 218
7.2 Asymptotics of Solutions 220
7.2.1 Description of the class of equations 220
7.2.2 The asymptotic expansion (the first case) 221
7.2.3 The asymptotic expansion (the second case) 222
7.2.4 /1 differential equations 221
7.2.5 The solution of nonhomogeneous equations 225
7.3 Deformations of Integral Transforms and Equations 229
7.3.1 General theory 229
7.3.2 Examples 231
10 Contents
8 Finiteness Theorems 243
8.1 Function Spaces 243
8.1.1 Preliminaries 244
8.1.2 Resurgent representations 245
8.1.3 Scales of function spaces 248
8.2 Spaces with Asymptotics 250
8.2.1 Preliminary considerations 250
8.2.2 Main definitions 251
8.2.3 Elements with simple singularities 254
8.3 Operator Algebras 255
8.3.1 The description of generators 255
8.3.2 Functions of generators 256
8.3.3 Construction of the operator algebras 258
8.3.4 Ellipticity and regularizers 260
8.4 The Finiteness Theorem for Differential Equations on Manifolds with
Cuspidal Points 262
8.4.1 Statement of the problem 262
8.4.2 The construction of local regularizers 265
8.4.3 The global regularizer and the Fredholm property 266
8.5 Asymptotic Expansions of Solutions 267
8.5.1 A preliminary transformation 268
8.5.2 A priori properties of the Borel transform of a solution 269
8.5.3 Multiplication by a function in the Borel representation .... 270
8.5.4 Proof of the theorem on endless continuability 274
8.6 Deformation of Resurgent Transforms 275
8.6.1 Definition of the deformation 275
8.6.2 Deformation of the operators and the index theorem 278
III Hyperbolic Equations 281
9 Equations of Borel Fuchs Type 283
9.1 Solution of the Problem in the Small 284
9.1.1 Degeneration of the first degree 285
9.1.2 Degeneration of higher degree 290
9.2 Solution of the Problem in the Large 296
9.2.1 Degeneracy of the first degree 296
9.2.2 Degeneracy of higher degree 300
9.3 Nonstationary problems in abstract algebras 302
9.3.1 General theory 302
9.3.2 Example 308
Contents 11
10 Vibration of Elastic Shells with Conical Points 313
10.1 The Model Example 314
10.1.1 Reduction of the problem 315
10.1.2 Asymptotic expansions of solutions 316
10.2 The General Case 330
10.2.1 The case of higher multiplicities 330
10.2.2 The multidimensional case 332
Appendices 341
A S/d s Transform of Ramified Analytic Functions 343
A.I Definition and Main Properties 344
A. 1.1 Auxiliary statements: Feynman integrals 344
A.1.2 The Thorn theorem 348
A.1.3 Definition of the transform 351
A.1.4 Commutation formulas 354
A.2 Functions with Simple Singularities 355
A.2.1 Definitions 355
A.2.2 The main theorem 356
A.2.3 Proof of the main theorem 356
A.2.4 Calculation of the leading term 359
B Some Elements of Noncommutative Analysis 361
B.I Formal Arithmetics 362
B.I.I The index permutation formula 363
B.I.2 The commutation formula 364
B.I.3 The derivation formula 364
B.I.4 Higher order expansions 364
B.2 The Method of Ordered Representations 365
Bibliography 367
Index 373
|
| any_adam_object | 1 |
| author | Schulze, Bert-Wolfgang 1944- Sternin, Boris Ju. 1939- Šatalov, Viktor E. 1945- |
| author_GND | (DE-588)120484579 (DE-588)120484536 (DE-588)120484544 |
| author_facet | Schulze, Bert-Wolfgang 1944- Sternin, Boris Ju. 1939- Šatalov, Viktor E. 1945- |
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| illustrated | Illustrated |
| indexdate | 2024-07-09T18:24:07Z |
| institution | BVB |
| isbn | 3527400869 |
| language | German |
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| physical | 376 S. graph. Darst. |
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| publishDateSearch | 1998 |
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| publisher | Wiley-VCH |
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| spelling | Schulze, Bert-Wolfgang 1944- Verfasser (DE-588)120484579 aut Differential equations on singular manifolds semiclassical theory and operator algebras Bert-Wolfgang Schulze ; Boris Sternin ; Victor Shatalov 1. ed. Berlin [u.a.] Wiley-VCH 1998 376 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Mathematical topics 15 Singularities (Mathematics) Variétés (Mathématiques) Équations différentielles elliptiques Équations différentielles hyperboliques Differential equations, Elliptic Differential equations, Hyperbolic Manifolds (Mathematics) Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Mannigfaltigkeit (DE-588)4037379-4 gnd rswk-swf Quasiklassische Näherung (DE-588)4296820-3 gnd rswk-swf Singularität Mathematik (DE-588)4077459-4 gnd rswk-swf Operatoralgebra (DE-588)4129366-6 gnd rswk-swf Mannigfaltigkeit (DE-588)4037379-4 s Singularität Mathematik (DE-588)4077459-4 s Partielle Differentialgleichung (DE-588)4044779-0 s Quasiklassische Näherung (DE-588)4296820-3 s Operatoralgebra (DE-588)4129366-6 s DE-604 Sternin, Boris Ju. 1939- Verfasser (DE-588)120484536 aut Šatalov, Viktor E. 1945- Verfasser (DE-588)120484544 aut Mathematical topics 15 (DE-604)BV008671507 15 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008294196&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Schulze, Bert-Wolfgang 1944- Sternin, Boris Ju. 1939- Šatalov, Viktor E. 1945- Differential equations on singular manifolds semiclassical theory and operator algebras Mathematical topics Singularities (Mathematics) Variétés (Mathématiques) Équations différentielles elliptiques Équations différentielles hyperboliques Differential equations, Elliptic Differential equations, Hyperbolic Manifolds (Mathematics) Partielle Differentialgleichung (DE-588)4044779-0 gnd Mannigfaltigkeit (DE-588)4037379-4 gnd Quasiklassische Näherung (DE-588)4296820-3 gnd Singularität Mathematik (DE-588)4077459-4 gnd Operatoralgebra (DE-588)4129366-6 gnd |
| subject_GND | (DE-588)4044779-0 (DE-588)4037379-4 (DE-588)4296820-3 (DE-588)4077459-4 (DE-588)4129366-6 |
| title | Differential equations on singular manifolds semiclassical theory and operator algebras |
| title_auth | Differential equations on singular manifolds semiclassical theory and operator algebras |
| title_exact_search | Differential equations on singular manifolds semiclassical theory and operator algebras |
| title_full | Differential equations on singular manifolds semiclassical theory and operator algebras Bert-Wolfgang Schulze ; Boris Sternin ; Victor Shatalov |
| title_fullStr | Differential equations on singular manifolds semiclassical theory and operator algebras Bert-Wolfgang Schulze ; Boris Sternin ; Victor Shatalov |
| title_full_unstemmed | Differential equations on singular manifolds semiclassical theory and operator algebras Bert-Wolfgang Schulze ; Boris Sternin ; Victor Shatalov |
| title_short | Differential equations on singular manifolds |
| title_sort | differential equations on singular manifolds semiclassical theory and operator algebras |
| title_sub | semiclassical theory and operator algebras |
| topic | Singularities (Mathematics) Variétés (Mathématiques) Équations différentielles elliptiques Équations différentielles hyperboliques Differential equations, Elliptic Differential equations, Hyperbolic Manifolds (Mathematics) Partielle Differentialgleichung (DE-588)4044779-0 gnd Mannigfaltigkeit (DE-588)4037379-4 gnd Quasiklassische Näherung (DE-588)4296820-3 gnd Singularität Mathematik (DE-588)4077459-4 gnd Operatoralgebra (DE-588)4129366-6 gnd |
| topic_facet | Singularities (Mathematics) Variétés (Mathématiques) Équations différentielles elliptiques Équations différentielles hyperboliques Differential equations, Elliptic Differential equations, Hyperbolic Manifolds (Mathematics) Partielle Differentialgleichung Mannigfaltigkeit Quasiklassische Näherung Singularität Mathematik Operatoralgebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008294196&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV008671507 |
| work_keys_str_mv | AT schulzebertwolfgang differentialequationsonsingularmanifoldssemiclassicaltheoryandoperatoralgebras AT sterninborisju differentialequationsonsingularmanifoldssemiclassicaltheoryandoperatoralgebras AT satalovviktore differentialequationsonsingularmanifoldssemiclassicaltheoryandoperatoralgebras |