The d-bar Neumann Problem and Schrödinger Operators
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 to Bergman spaces of holomorphic d-bar functions in one and several complex variables.These operators...
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| Format: | E-Book |
| Sprache: | Englisch Deutsch |
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Germany
De Gruyter
2014
Walter de Gruyter GmbH |
| Ausgabe: | 1 |
| Schriftenreihe: | De Gruyter Expositions in Mathematics |
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| ISBN: | 9783110315356, 3110315351, 9783110315301, 3110315300 |
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| Abstract | 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 to Bergman spaces of holomorphic d-bar functions in one and several complex variables.These operators are Hankel operators of special type. |
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| AbstractList | 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 to Bergman spaces of holomorphic d-bar functions in one and several complex variables.These operators are Hankel operators of special type. The topic of this bookis located at the intersection of complex analysis, operator theory and partial differential equations. First we investigate the canonical solution operator to d-bar restricted to Bergman spaces of holomorphicL 2functions in one and several complex variables. These operators are Hankel operators of special type. In the following we consider the general d-bar-complex and derive properties of the complex Laplacian onL 2spaces of bounded pseudoconvex domains and on weightedL 2spaces. The main part is devoted to compactness of the d-bar-Neumann operator. The last part will contain a detailed account of the application of the d-bar-methods to Schrödinger operators, Pauli and Dirac operators and to Witten-Laplacians. 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. 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 ond-bar spaces over bounded pseudoconvex domains and on weightedd-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. |
| Author | Haslinger, Friedrich |
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| Keywords | Neumannproblem Inhomogeneous Cauchy-Riemann Equation Hankel Operator Schrödingeroperator Compactness d-bar Neumann Problem Schrödinger Operator Witten Laplacian Funktionentheorie |
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| PublicationDate | 2014 [2014] 2014-08-20 |
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| Snippet | 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... 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... The topic of this bookis located at the intersection of complex analysis, operator theory and partial differential equations. First we investigate the... |
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| SubjectTerms | Compactness d-bar Neumann Problem Funktionentheorie Hankel Operator Inhomogeneous Cauchy-Riemann Equation Mathematics MATHEMATICS / General Neumann problem Neumannproblem Schrödinger Operator Schrödingeroperator Witten Laplacian |
| TableOfContents | Intro -- Preface -- Contents -- 1 Bergman spaces -- 1.1 Elementary properties -- 1.2 Examples -- 1.3 Biholomorphic maps -- 1.4 Notes -- 2 The canonical solution operator to ?? -- 2.1 Compact operators on Hilbert spaces -- 2.2 The canonical solution operator to ∂̄ restricted to A2(D) -- 2.3 Notes -- 3 Spectral properties of the canonical solution operator to -- 3.1 Complex differential forms -- 3.2 (0, 1)-forms with holomorphic coefficients -- 3.3 Compactness and Schatten class membership -- 3.4 Notes -- 4 The ∂̄ -complex -- 4.1 Unbounded operators on Hilbert spaces -- 4.2 Distributions -- 4.3 A finite-dimensional analog -- 4.4 The ∂̄ -Neumann operator -- 4.5 Notes -- 5 Density of smooth forms -- 5.1 Friedrichs' Lemma and Sobolev spaces -- 5.2 Density in the graph norm -- 5.3 Notes -- 6 The weighted ∂̄-complex -- 6.1 The ∂̄-Neumann operator on (0, 1)-forms -- 6.2 (0, q)-forms -- 6.3 Notes -- 7 The twisted ∂̄-complex -- 7.1 An exact sequence of unbounded operators -- 7.2 The twisted basic estimates -- 7.3 Notes -- 8 Applications -- 8.1 Hörmander's L2-estimates -- 8.2 Weighted spaces of entire functions -- 8.3 Notes -- 9 Spectral analysis -- 9.1 Resolutions of the identity -- 9.2 Spectral decomposition of bounded normal operators -- 9.3 Spectral decomposition of unbounded self-adjoint operators -- 9.4 Determination of the spectrum -- 9.5 Variational characterization of the discrete spectrum -- 9.6 Notes -- 10 Schrödinger operators and Witten-Laplacians -- 10.1 Difference quotients -- 10.2 Interior regularity -- 10.3 Schrödinger operators with magnetic field -- 10.4 Witten-Laplacians -- 10.5 Dirac and Pauli operators -- 10.6 Notes -- 11 Compactness -- 11.1 Precompact sets in L2-spaces -- 11.2 Sobolev spaces and Gårding's inequality -- 11.3 Compactness in weighted spaces -- 11.4 Bounded pseudoconvex domains -- 11.5 Notes 12 The ∂̄-Neumann operator and the Bergman projection -- 12.1 The Stone-Weierstraß Theorem -- 12.2 Commutators of the Bergman projection -- 12.3 Notes -- 13 Compact resolvents -- 13.1 Schrödinger operators -- 13.2 Dirac and Pauli operators -- 13.3 Notes -- 14 Spectrum of ⃞ on the Fock space -- 14.1 The general setting -- 14.2 Determination of the spectrum -- 14.3 Notes -- 15 Obstructions to compactness -- 15.1 The bidisc -- 15.2 Weighted spaces -- 15.3 Notes -- Bibliography -- Index 2. The canonical solution operator to ∂̄ 3. Spectral properties of the canonical solution operator to ∂̄ 11. Compactness 13. Compact resolvents 5. Density of smooth forms Index 7. The twisted ∂̄-complex 14. Spectrum of ◻ on the Fock space - 15. Obstructions to compactness 8. Applications 10. Schrödinger operators and Witten–Laplacians 6. The weighted ∂̄-complex Backmatter Contents 12. The ∂̄-Neumann operator and the Bergman projection 9. Spectral analysis Frontmatter -- 1. Bergman spaces Preface Bibliography 4. The ∂̄-complex |
| Title | The d-bar Neumann Problem and Schrödinger Operators |
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