Ground state energy of the magnetic Laplacian on corner domains:
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Paris
Société Mathématique de France
2016
|
| Schriftenreihe: | Mémoires de la SMF
145 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | Literaturverzeichnis Seite [129]-135 |
| Beschreibung: | vii, 138 Seiten |
| ISBN: | 9782856298305 |
Internformat
MARC
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| 100 | 1 | |a Bonnaillie-Noël, Virginie |d 1976- |0 (DE-588)1097243001 |4 aut | |
| 245 | 1 | 0 | |a Ground state energy of the magnetic Laplacian on corner domains |c Virginie Bonnaillie-Noël, Monique Dauge, Nicolas Popoff |
| 264 | 1 | |a Paris |b Société Mathématique de France |c 2016 | |
| 300 | |a vii, 138 Seiten | ||
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| 490 | 1 | |a Mémoires de la SMF |v 145 | |
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Datensatz im Suchindex
| _version_ | 1804176160719896576 |
|---|---|
| adam_text | CONTENTS
R
Part I. Introduction ..................................................... 1
1. Introduction of the problem and main results .......................... 3
1.1. The magnetic Laplacian and its lowest eigenvalue .................... 4
1.2. Local ground state energies ......................................... 6
1.3. Asymptotic formulas with remainders ................................. 8
1.4. Contents ........................................................... 11
1.5. Notations .......................................................... 13
2. State of the art ...................................................... 15
2.1. Without boundary or with Dirichlet conditions ...................... 15
2.2. Neumann conditions in dimension 2 ............................ 16
2.3. Neumann conditions in dimension 3 ............................ 18
Part II. Corner structure and lower bounds ............................... 21
3. Domains with corners and their singular chains ........................ 23
3.1. Tangent cones and corner domains ................................... 23
3.2. Admissible atlases ................................................. 27
3.3. Estimates for local Jacobian matrices .............................. 30
3.4. Strata and singular chains ......................................... 34
3.5. 3D domains ......................................................... 41
4. Magnetic Laplacians and their tangent operators ....................... 45
4.1. Change of variables ............................................... 45
4.2. Model and tangent operators ....................................... 46
4.3. Linearization ...................................................... 47
4.4. A genera] rough upper bound ........................................ 49
5. Lower bounds for ground state energy in corner domains ................ 53
5.1. Estimates outside conical points ................................... 54
vi
CONTENTS
5.2. Estimates near conical points .................................... 56
5.3. Generalization.................................................... 59
Part III. Upper bounds .................................................. 61
6. Taxonomy of model problems ........................................... 63
6.1. Full space (d = 0) ............................................... 64
6.2. Half-space (d = 1) ............................................... 64
6.3. Wedges (d = 2) ................................................... 66
6.4. 3D cones (d = 3) ................................................. 67
7. Dichotomy and substructures for model problems ....................... 69
7.1. Admissible Generalized Eigenvectors .............................. 69
7.2. Dichotomy Theorem ................................................ 70
7.3. Examples ......................................................... 73
7.4. Scaling and truncating Admissible Generalized Eigenvectors ....... 74
8. Properties of the local ground state energy .......................... 77
8.1. Lower semicontinuity ............................................ 77
8.2. Positivity of the ground state energy ........................... 78
9. Upper bounds for ground state energy in corner domains ............... 81
9.1. Principles of construction for quasimodes ........................ 82
9.2. First level of construction and sitting quasimodes ............... 84
9.3. Second level of construction and sliding quasimodes .............. 87
9.4. Third level of construction and doubly sliding quasimodes ........ 90
9.5. Conclusion ....................................................... 91
Part IV. Improved upper bounds .......................................... 93
10. Stability of Admissible Generalized Eigenvectors .................... 95
10.1. Structure of AGE’s ............................................. 95
10.2. Stability under perturbation ................................... 96
11. Improvement of upper bounds for more regular magnetic fields . . 99
11.1. (Gl) One direction of exponential decay ........................100
11.2. (G2) Two directions of exponential decay .......................107
12. Conclusion: Improvements and extensions ..............................113
12.1. Corner concentration and standard consequences .................113
12.2. The necessity of a taxonomy ....................................114
12.3. Continuity of local energies.....................................114
12.4. Dirichlet boundary conditions .................................. 115
12.5. Robin boundary conditions with a large parameter for the Laplacian . . 116
MEMOIRES DE LA SMF 145
CONTENTS vii
Part V. Appendices ....................................................117
A. Magnetic identities ...............................................119
A.l. Gauge transform ...............................................119
A.2. Change of variables ...........................................120
A.3. Comparison formula.............................................121
A.4. Cut-off effect ................................................121
B. Partition of unity suitable for IMS type formulas .................123
Bibliography ..........................................................129
Index .................................................................137
SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2016
The asymptotic behavior of the first eigenvalue of a magnetic Laplacian in the strong field limit and with the von Neumann realization in a smooth domain is characterized for dimensions 2 and 3 by model problems inside the domain or on its boundary. In dimension 2, for polygonal domains, a new set of model problems on sectors has to be taken into account. In this work, we consider the class of general corner domains. In dimension 3, they include as particular cases polyhedra and axisymmetric cones. We attach model problems not only to each point of the closure of the domain, but also to a hierarchy of “tangent substructures” associated with singular chains. We investigate spectral properties of these model problems, namely semicontinuity and existence of bounded generalized eigenfunctions. We prove estimates for the remainders of our asymptotic formula. Lower bounds are obtained with the help of an IMS type partition based on adequate two-scale coverings of the corner domain, whereas upper bounds are established by a novel construction of quasimodes, qualified as sitting or sliding according to spectral properties of local model problems. A part of our analysis extends to any dimension.
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| genre_facet | Monografische Reihe |
| id | DE-604.BV043516668 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T07:27:46Z |
| institution | BVB |
| isbn | 9782856298305 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-028932704 |
| oclc_num | 947132399 |
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| physical | vii, 138 Seiten |
| publishDate | 2016 |
| publishDateSearch | 2016 |
| publishDateSort | 2016 |
| publisher | Société Mathématique de France |
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| series | Mémoires de la SMF |
| series2 | Mémoires de la SMF |
| spelling | Bonnaillie-Noël, Virginie 1976- (DE-588)1097243001 aut Ground state energy of the magnetic Laplacian on corner domains Virginie Bonnaillie-Noël, Monique Dauge, Nicolas Popoff Paris Société Mathématique de France 2016 vii, 138 Seiten txt rdacontent n rdamedia nc rdacarrier Mémoires de la SMF 145 Literaturverzeichnis Seite [129]-135 Quantentheorie (DE-588)4047992-4 gnd rswk-swf Hamilton-Operator (DE-588)4072278-8 gnd rswk-swf (DE-588)4179998-7 Monografische Reihe gnd-content Quantentheorie (DE-588)4047992-4 s Hamilton-Operator (DE-588)4072278-8 s DE-604 Dauge, Monique aut Popoff, Nicolas aut Mémoires de la SMF 145 (DE-604)BV000000921 145 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028932704&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028932704&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Bonnaillie-Noël, Virginie 1976- Dauge, Monique Popoff, Nicolas Ground state energy of the magnetic Laplacian on corner domains Mémoires de la SMF Quantentheorie (DE-588)4047992-4 gnd Hamilton-Operator (DE-588)4072278-8 gnd |
| subject_GND | (DE-588)4047992-4 (DE-588)4072278-8 (DE-588)4179998-7 |
| title | Ground state energy of the magnetic Laplacian on corner domains |
| title_auth | Ground state energy of the magnetic Laplacian on corner domains |
| title_exact_search | Ground state energy of the magnetic Laplacian on corner domains |
| title_full | Ground state energy of the magnetic Laplacian on corner domains Virginie Bonnaillie-Noël, Monique Dauge, Nicolas Popoff |
| title_fullStr | Ground state energy of the magnetic Laplacian on corner domains Virginie Bonnaillie-Noël, Monique Dauge, Nicolas Popoff |
| title_full_unstemmed | Ground state energy of the magnetic Laplacian on corner domains Virginie Bonnaillie-Noël, Monique Dauge, Nicolas Popoff |
| title_short | Ground state energy of the magnetic Laplacian on corner domains |
| title_sort | ground state energy of the magnetic laplacian on corner domains |
| topic | Quantentheorie (DE-588)4047992-4 gnd Hamilton-Operator (DE-588)4072278-8 gnd |
| topic_facet | Quantentheorie Hamilton-Operator Monografische Reihe |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028932704&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028932704&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000921 |
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