Elliptic Boundary Problems for Dirac Operators

1. Front Matter

   Pages i-xviii

   PDF

2. Clifford Algebras and Dirac Operators

   1. Front Matter

      Pages 1-1

      PDF

   2. Clifford Algebras and Clifford Modules

      • Bernhelm Booß-Bavnbek, Krzysztof P. Wojciechowski

      Pages 3-9

   3. Clifford Bundles and Compatible Connections

      • Bernhelm Booß-Bavnbek, Krzysztof P. Wojciechowski

      Pages 10-18

   4. Dirac Operators

      • Bernhelm Booß-Bavnbek, Krzysztof P. Wojciechowski

      Pages 19-25

   5. Dirac Laplacian and Connection Laplacian

      • Bernhelm Booß-Bavnbek, Krzysztof P. Wojciechowski

      Pages 26-28

   6. Euclidean Examples

      • Bernhelm Booß-Bavnbek, Krzysztof P. Wojciechowski

      Pages 29-35

   7. The Classical Dirac (Atiyah-Singer) Operators on Spin Manifolds

      • Bernhelm Booß-Bavnbek, Krzysztof P. Wojciechowski

      Pages 36-39

   8. Dirac Operators and Chirality

      • Bernhelm Booß-Bavnbek, Krzysztof P. Wojciechowski

      Pages 40-42

   9. Unique Continuation Property for Dirac Operators

      • Bernhelm Booß-Bavnbek, Krzysztof P. Wojciechowski

      Pages 43-49

   10. Invertible Doubles

       • Bernhelm Booß-Bavnbek, Krzysztof P. Wojciechowski

       Pages 50-58

   11. Glueing Constructions. Relative Index Theorem

       • Bernhelm Booß-Bavnbek, Krzysztof P. Wojciechowski

       Pages 59-63

3. Analytical and Topological Tools

   1. Front Matter

      Pages 65-65

      PDF

   2. Sobolev Spaces on Manifolds with Boundary

      • Bernhelm Booß-Bavnbek, Krzysztof P. Wojciechowski

      Pages 67-74

   3. Calderón Projector for Dirac Operators

      • Bernhelm Booß-Bavnbek, Krzysztof P. Wojciechowski

      Pages 75-94

   4. Existence of Traces of Null Space Elements

      • Bernhelm Booß-Bavnbek, Krzysztof P. Wojciechowski

      Pages 95-104

   5. Spectral Projections of Dirac Operators

      • Bernhelm Booß-Bavnbek, Krzysztof P. Wojciechowski

      Pages 105-110

   6. Pseudo-Differential Grassmannians

      • Bernhelm Booß-Bavnbek, Krzysztof P. Wojciechowski

      Pages 111-126

   7. The Homotopy Groups of the Space of Self-Adjoint Fredholm Operators

      • Bernhelm Booß-Bavnbek, Krzysztof P. Wojciechowski

      Pages 127-137

   8. The Spectral Flow of Families of Self-Adjoint Operators

      • Bernhelm Booß-Bavnbek, Krzysztof P. Wojciechowski

      Pages 138-160

Elliptic boundary problems have enjoyed interest recently, espe­ cially among C* -algebraists and mathematical physicists who want to understand single aspects of the theory, such as the behaviour of Dirac operators and their solution spaces in the case of a non-trivial boundary. However, the theory of elliptic boundary problems by far has not achieved the same status as the theory of elliptic operators on closed (compact, without boundary) manifolds. The latter is nowadays rec­ ognized by many as a mathematical work of art and a very useful technical tool with applications to a multitude of mathematical con­ texts. Therefore, the theory of elliptic operators on closed manifolds is well-known not only to a small group of specialists in partial dif­ ferential equations, but also to a broad range of researchers who have specialized in other mathematical topics. Why is the theory of elliptic boundary problems, compared to that on closed manifolds, still lagging behind in popularity? Admittedly, from an analytical point of view, it is a jigsaw puzzle which has more pieces than does the elliptic theory on closed manifolds. But that is not the only reason.

• Manifold
• Sobolev space
• algebra
• equation
• theorem
• partial differential equations
• matrix theory
• ordinary differential equations

• IMFUFA, Roskilde University, Roskilde, Denmark

  Bernhelm Booß-Bavnbek

• Department of Mathematics, IUPUI, Indianapolis, USA

  Krzysztof P. Wojciechowski

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