Haslinger The d-bar Neumann Problem and Schrödinger Operators

1. Auflage 2014
ISBN: 978-3-11-031535-6
Verlag: De Gruyter
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, Band 59, 252 Seiten

Reihe: De Gruyter Expositions in MathematicsISSN

The topic of this book is located at the intersection of complex analysis, operator theory and partial differential equations. It begins with results on the canonical solution operator to restricted toBergman spaces of holomorphic d-bar functions in one and several complex variables.These operators are Hankel operators of special type. In the following the general complex is investigated on d-bar spaces over bounded pseudoconvex domains and on weighted d-bar spaces. The main part is devoted to the spectral analysis of the complex Laplacian and to compactness of the Neumann operator.The last part contains a detailed account of the application of the methods to Schrödinger operators, Pauli and Dirac operators and to Witten-Laplacians. It is assumed that the reader has a basic knowledge of complex analysis, functional analysis and topology. With minimal prerequisites required, this book provides a systematic introduction to an active area of research for both students at a bachelor level and mathematicians.

Zielgruppe

Graduate students and scientists in complex analysis, operator theory and partial differential equations; Academic libraries

Autoren/Hrsg.

Haslinger, Friedrich

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Funktionentheorie, Komplexe Analysis
• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen

Inhaltsverzeichnis

1;Preface;7
2;Contents;9
3;1 Bergman spaces;13
3.1;1.1 Elementary properties;13
3.2;1.2 Examples;20
3.3;1.3 Biholomorphic maps;24
3.4;1.4 Notes;27
4;2 The canonical solution operator to ??;28
4.1;2.1 Compact operators on Hilbert spaces;28
4.2;2.2 The canonical solution operator to
.¯ restricted to A2(D);38
4.3;2.3 Notes;47
5;3 Spectral properties of the canonical solution operator to
;48
5.1;3.1 Complex differential forms;48
5.2;3.2 (0, 1)-forms with holomorphic coefficients;49
5.3;3.3 Compactness and Schatten class membership;51
5.4;3.4 Notes;61
6;4 The .¯ -complex
;62
6.1;4.1 Unbounded operators on Hilbert spaces;62
6.2;4.2 Distributions;77
6.3;4.3 A finite-dimensional analog;83
6.4;4.4 The .¯ -Neumann operator
;84
6.5;4.5 Notes;98
7;5 Density of smooth forms;99
7.1;5.1 Friedrichs’ Lemma and Sobolev spaces;99
7.2;5.2 Density in the graph norm;110
7.3;5.3 Notes;115
8;6 The weighted .¯-complex ;116
8.1;6.1 The .¯-Neumann operator
on (0, 1)-forms;116
8.2;6.2 (0, q)-forms
;121
8.3;6.3 Notes;124
9;7 The twisted .¯-complex
;126
9.1;7.1 An exact sequence of unbounded operators;126
9.2;7.2 The twisted basic estimates;127
9.3;7.3 Notes;130
10;8 Applications;131
10.1;8.1 Hörmander’s L2-estimates
;131
10.2;8.2 Weighted spaces of entire functions;134
10.3;8.3 Notes;138
11;9 Spectral analysis;139
11.1;9.1 Resolutions of the identity;139
11.2;9.2 Spectral decomposition of bounded normal operators;142
11.3;9.3 Spectral decomposition of unbounded self-adjoint operators;148
11.4;9.4 Determination of the spectrum;162
11.5;9.5 Variational characterization of the discrete spectrum;172
11.6;9.6 Notes;177
12;10 Schrödinger operators and Witten–Laplacians;178
12.1;10.1 Difference quotients;178
12.2;10.2 Interior regularity;180
12.3;10.3 Schrödinger operators with magnetic field;183
12.4;10.4 Witten–Laplacians;190
12.5;10.5 Dirac and Pauli operators;192
12.6;10.6 Notes;194
13;11 Compactness;195
13.1;11.1 Precompact sets in L2-spaces
;195
13.2;11.2 Sobolev spaces and Gårding’s inequality;198
13.3;11.3 Compactness in weighted spaces;202
13.4;11.4 Bounded pseudoconvex domains;214
13.5;11.5 Notes;217
14;12 The .¯-Neumann
operator and the Bergman projection;219
14.1;12.1 The Stone–Weierstraß Theorem;219
14.2;12.2 Commutators of the Bergman projection;222
14.3;12.3 Notes;227
15;13 Compact resolvents;228
15.1;13.1 Schrödinger operators;228
15.2;13.2 Dirac and Pauli operators;230
15.3;13.3 Notes;232
16;14 Spectrum of .
on the Fock space;233
16.1;14.1 The general setting;233
16.2;14.2 Determination of the spectrum;235
16.3;14.3 Notes;240
17;15 Obstructions to compactness;241
17.1;15.1 The bidisc;241
17.2;15.2 Weighted spaces;242
17.3;15.3 Notes;246
18;Bibliography;247
19;Index;251

Über Autor(innen)

Friedrich Haslinger, University of Vienna, Austria.