Singular perturbations of differential operators: solvable Schrödinger type operators
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
2000
|
| Edition: | 1. publ. |
| Series: | London Mathematical Society: London Mathematical Society lecture note series
271 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 429 S. |
| ISBN: | 052177912X |
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MARC
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| 100 | 1 | |a Albeverio, Sergio |d 1939- |e Verfasser |0 (DE-588)121093999 |4 aut | |
| 245 | 1 | 0 | |a Singular perturbations of differential operators |b solvable Schrödinger type operators |c S. Albeverio ; P. Kurasov |
| 246 | 1 | 3 | |a m |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2000 | |
| 300 | |a XIV, 429 S. | ||
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| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a London Mathematical Society: London Mathematical Society lecture note series |v 271 | |
| 650 | 4 | |a Operador de Schrödinger | |
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| 650 | 0 | 7 | |a Hamilton-Operator |0 (DE-588)4072278-8 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Singularität |g Mathematik |0 (DE-588)4077459-4 |2 gnd |9 rswk-swf |
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Record in the Search Index
| _version_ | 1804127443381911552 |
|---|---|
| adam_text | Contents
1 Rank one perturbations 9
1.1 Bounded perturbations 9
1.1.1 Resolvent analysis 9
1.1.2 Infinite coupling 14
1.2 Krein s formula 15
1.2.1 Bounded and singular perturbations 15
1.2.2 Scale of Hilbert spaces 17
1.2.3 Form bounded and form unbounded
perturbations 19
1.2.4 Rank one perturbations and the extension
theory for S3rmmetric operators 21
1.2.5 The extension theory and Krein s formula 23
1.2.6 Q function for rank one perturbations 28
1.3 Singular rank one perturbations 30
1.3.1 Form bounded rank one singular perturbations 30
1.3.2 Family of rank one form unbounded
perturbations 32
1.3.3 Singular rank one form unbounded perturbations of
homogeneous operators 35
1.3.4 Resolvent formulas 38
1.4 Approximations of singular rank one perturbations 41
1.4.1 Norm convergence of the approximations 41
1.4.2 Strong resolvent convergence of the
approximations 46
1.5 Differential operators with rank one singular perturbations . . 49
1.5.1 Point interactions in dimension three 49
1.5.2 Perturbations of the first derivative operator 58
1.5.3 Dirac operator with a pseudopotential 60
xi
xii CONTENTS
2 Generalized rank one perturbations 63
2.1 Krein s formula for the generalized resolvents 63
2.1.1 Generalized self adjoint extensions and
generalized resolvents 63
2.1.2 Generalized rank one perturbations 64
2.2 Model generalized perturbations 69
2.2.1 Introduction 69
2.2.2 Model perturbations I: densely defined restricted oper¬
ators 70
2.2.3 Model perturbations II: non densely defined
construction 81
2.2.4 Generalized resolvents and model generalized
perturbations 90
2.3 Generalized point interactions 92
2.3.1 Generalized point interaction in dimension three .... 92
2.3.2 Generalized delta interaction in dimension one 99
3 Finite rank perturbations and distribution theory 111
3.1 Finite rank perturbations Ill
3.1.1 Preliminaries Ill
3.1.2 Form bounded finite rank perturbations 116
3.1.3 Form unbounded finite rank perturbations 120
3.1.4 Generalized finite rank perturbations 125
3.2 Point interactions for differential operators and distribution
theory 130
3.2.1 Point interactions for differential operators as
finite rank perturbations 130
3.2.2 Distribution theory for discontinuous test functions . . 134
3.2.3 Differential operator of order n in one
dimension 140
3.2.4 Second order differential operator in one
dimension 142
4 Scattering theory for finite rank perturbations 159
4.1 Scattering theory for rank one perturbations 159
4.1.1 Rank one perturbations and operators with a simple
spectrum 160
4.1.2 Invariance of the absolutely continuous
spectrum 162
4.1.3 Wave operators and scattering operator
for rank one perturbations 169
CONTENTS xiii
4.2 Scattering for self adjoint extensions 175
4.2.1 Wave operators for self adjoint extensions 175
4.2.2 Scattering operator for self adjoint extensions 177
4.2.3 Scattering matrix for self adjoint extensions 184
4.3 Scattering theory for finite rank perturbations 188
4.3.1 Scattering theory for finite rank perturbations 188
4.3.2 Scattering matrix for rank two perturbations 190
4.3.3 Scattering matrix for generalized perturbations .... 191
5 Krein s formula for infinite deficiency indices and
two body problems 195
5.1 Infinite rank perturbations 195
5.1.1 Krein s formula for infinite deficiency indices 195
5.1.2 Generalized perturbations of infinite rank 198
5.1.3 Resolvent formula for functionals 202
5.2 Two body problems 205
5.2.1 Two body operator with interaction of rank one .... 205
5.2.2 Two body operator with generalized interaction of rank
one 219
6 Few body problems 227
6.1 Few body formula I: self adjoint extensions 227
6.1.1 Tensor structure of the few body Hilbert space .... 227
6.1.2 Few body operator with H i interaction 232
6.1.3 Few body operator with H 2 interaction 238
6.1.4 Self adjointness of the few body operator
with strongly separable interactions 244
6.1.5 Few body operator with delta interaction 247
6.2 Few body formula II: generalized self adjoint extensions .... 248
6.2.1 Generalized unperturbed operator 248
6.2.2 Inner cluster generalized interaction 250
6.2.3 Few body operator with generalized interaction .... 256
6.2.4 Self adjointness of the few body operator with infinites
imally separable generalized interactions 265
7 Three body models in one dimension 273
7.1 Schrodinger operators constructed using self adjoint extensions 273
7.1.1 Tensor structure of the Hilbert space 273
7.1.2 Definition of the Schrodinger operator as
a self adjoint operator 274
7.1.3 Spectrum, eigenfunctions and Bethe Ansatz 277
xiv CONTENTS
7.2 Operators with generalized delta interactions 281
7.2.1 Two body generalized delta interactions 281
7.2.2 Three body Schrodinger operator with two body gen¬
eralized delta interactions 283
7.2.3 Symmetry group 285
7.2.4 Outgoing wave and Sommerfeld Maluzhinetz
transformation 293
7.2.5 Solution of the functional equations 298
7.2.6 Properties of the outgoing wave 305
7.2.7 Spectrum and scattering matrix 315
7.2.8 Calculation of the scattering matrix 318
A Historical remarks. 327
A.I Extension theory for symmetric
operators 327
A.I.I Extension theory and Nevanlinna /^ functions 327
A. 1.2 Symplectic structure and von Neumann construction . 330
A.2 Finite rank perturbations 332
A.2.1 Definition of singular finite rank perturbations 332
A.2.2 Generalized singular perturbations 334
A.2.3 Singular perturbations defined by forms 336
A.3 Point interactions 337
A.3.1 Definition of the point interactions in RJ,R2 and R3 . 337
A.3.2 Approximations of point interactions 340
A.3.3 Point interactions and inverse problems 341
A.4 Few body problems 341
A.4.1 Three body problems in R3 and R2 341
A.4.2 Few body problems in R1 344
A.5 Recent developments 346
A.5.1 Sphere interactions 346
A.5.2 Interactions on low dimensional manifolds 346
A.5.3 Relativistic point interactions 347
A.5.4 Self adjoint operators on graphs 348
A.5.5 Acoustic problems 349
A.5.6 Magnetic field, Aharonov Bohm effect and
anyons 349
A.5.7 Time dependent interactions 350
A.5.8 Solid state 350
A.5.9 Further developments 351
Bibliography 353
Index 427
|
| any_adam_object | 1 |
| author | Albeverio, Sergio 1939- Kurasov, Pavel |
| author_GND | (DE-588)121093999 |
| author_facet | Albeverio, Sergio 1939- Kurasov, Pavel |
| author_role | aut aut |
| author_sort | Albeverio, Sergio 1939- |
| author_variant | s a sa p k pk |
| building | Verbundindex |
| bvnumber | BV012774367 |
| classification_rvk | SI 320 SK 620 |
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| dewey-full | 515.7242 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.7242 |
| dewey-search | 515.7242 |
| dewey-sort | 3515.7242 |
| dewey-tens | 510 - Mathematics |
| discipline | Physik Mathematik |
| edition | 1. publ. |
| format | Book |
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| isbn | 052177912X |
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| spelling | Albeverio, Sergio 1939- Verfasser (DE-588)121093999 aut Singular perturbations of differential operators solvable Schrödinger type operators S. Albeverio ; P. Kurasov m 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2000 XIV, 429 S. txt rdacontent n rdamedia nc rdacarrier London Mathematical Society: London Mathematical Society lecture note series 271 Operador de Schrödinger Perturbación (Matemáticas) Hamilton-Operator (DE-588)4072278-8 gnd rswk-swf Singularität Mathematik (DE-588)4077459-4 gnd rswk-swf Störungstheorie (DE-588)4128420-3 gnd rswk-swf (DE-588)4113937-9 Hochschulschrift gnd-content Hamilton-Operator (DE-588)4072278-8 s Störungstheorie (DE-588)4128420-3 s Singularität Mathematik (DE-588)4077459-4 s DE-604 Kurasov, Pavel Verfasser aut London Mathematical Society: London Mathematical Society lecture note series 271 (DE-604)BV000000130 271 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008687049&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Albeverio, Sergio 1939- Kurasov, Pavel Singular perturbations of differential operators solvable Schrödinger type operators London Mathematical Society: London Mathematical Society lecture note series Operador de Schrödinger Perturbación (Matemáticas) Hamilton-Operator (DE-588)4072278-8 gnd Singularität Mathematik (DE-588)4077459-4 gnd Störungstheorie (DE-588)4128420-3 gnd |
| subject_GND | (DE-588)4072278-8 (DE-588)4077459-4 (DE-588)4128420-3 (DE-588)4113937-9 |
| title | Singular perturbations of differential operators solvable Schrödinger type operators |
| title_alt | m |
| title_auth | Singular perturbations of differential operators solvable Schrödinger type operators |
| title_exact_search | Singular perturbations of differential operators solvable Schrödinger type operators |
| title_full | Singular perturbations of differential operators solvable Schrödinger type operators S. Albeverio ; P. Kurasov |
| title_fullStr | Singular perturbations of differential operators solvable Schrödinger type operators S. Albeverio ; P. Kurasov |
| title_full_unstemmed | Singular perturbations of differential operators solvable Schrödinger type operators S. Albeverio ; P. Kurasov |
| title_short | Singular perturbations of differential operators |
| title_sort | singular perturbations of differential operators solvable schrodinger type operators |
| title_sub | solvable Schrödinger type operators |
| topic | Operador de Schrödinger Perturbación (Matemáticas) Hamilton-Operator (DE-588)4072278-8 gnd Singularität Mathematik (DE-588)4077459-4 gnd Störungstheorie (DE-588)4128420-3 gnd |
| topic_facet | Operador de Schrödinger Perturbación (Matemáticas) Hamilton-Operator Singularität Mathematik Störungstheorie Hochschulschrift |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008687049&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000130 |
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