The Callias Index Formula Revisited

  • Fritz Gesztesy
  • Marcus Waurick

Part of the Lecture Notes in Mathematics book series (LNM, volume 2157)

Table of contents

  1. Front Matter
    Pages i-ix
  2. Fritz Gesztesy, Marcus Waurick
    Pages 1-8
  3. Fritz Gesztesy, Marcus Waurick
    Pages 9-11
  4. Fritz Gesztesy, Marcus Waurick
    Pages 13-21
  5. Fritz Gesztesy, Marcus Waurick
    Pages 23-33
  6. Fritz Gesztesy, Marcus Waurick
    Pages 35-53
  7. Fritz Gesztesy, Marcus Waurick
    Pages 55-63
  8. Fritz Gesztesy, Marcus Waurick
    Pages 65-76
  9. Fritz Gesztesy, Marcus Waurick
    Pages 77-99
  10. Fritz Gesztesy, Marcus Waurick
    Pages 101-105
  11. Fritz Gesztesy, Marcus Waurick
    Pages 107-117
  12. Fritz Gesztesy, Marcus Waurick
    Pages 119-129
  13. Fritz Gesztesy, Marcus Waurick
    Pages 131-150
  14. Fritz Gesztesy, Marcus Waurick
    Pages 151-156
  15. Back Matter
    Pages 167-194

About this book

Introduction

These lecture notes aim at providing a purely analytical and accessible proof of the Callias index formula. In various branches of mathematics (particularly, linear and nonlinear partial differential operators, singular integral operators, etc.) and theoretical physics (e.g., nonrelativistic and relativistic quantum mechanics, condensed matter physics, and quantum field theory), there is much interest in computing Fredholm indices of certain linear partial differential operators. In the late 1970’s, Constantine Callias found a formula for the Fredholm index of a particular first-order differential operator (intimately connected to a supersymmetric Dirac-type operator) additively perturbed by a potential, shedding additional light on the Fedosov-Hörmander Index Theorem. As a byproduct of our proof we also offer a glimpse at special non-Fredholm situations employing a generalized Witten index.

Keywords

47A53,47F05,47B25 Callias Index Formula Dirac-type Operator Witten Index Fredholm Theory Trace Class Operators

Authors and affiliations

  • Fritz Gesztesy
    • 1
  • Marcus Waurick
    • 2
  1. 1.Dept of MathematicsUniversity of MissouriColumbiaUSA
  2. 2.Institut für AnalysisTU DresdenDresdenGermany

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-29977-8
  • Copyright Information Springer International Publishing Switzerland 2016
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-29976-1
  • Online ISBN 978-3-319-29977-8
  • Series Print ISSN 0075-8434
  • Series Online ISSN 1617-9692
  • About this book