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: | eBook |
| Language: | English German |
| Published: |
Germany
De Gruyter
2014
Walter de Gruyter GmbH |
| Edition: | 1 |
| Series: | De Gruyter Expositions in Mathematics |
| Subjects: | |
| ISBN: | 9783110315356, 3110315351, 9783110315301, 3110315300 |
| Online Access: | Get full text |
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Table of Contents:
- 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