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  • © 2016

The Callias Index Formula Revisited

  • Offers a new, functional analytic approach to the Callias index theorem and generalises it considerably
  • Give a very detailed history and explain the background in great detail
  • Shows very clearly the connections with other areas throughout the manuscript
  • Includes supplementary material: sn.pub/extras

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

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Table of contents (14 chapters)

  1. Front Matter

    Pages i-ix
  2. Introduction

    • Fritz Gesztesy, Marcus Waurick
    Pages 1-8
  3. Notational Conventions

    • Fritz Gesztesy, Marcus Waurick
    Pages 9-11
  4. Functional Analytic Preliminaries

    • Fritz Gesztesy, Marcus Waurick
    Pages 13-21
  5. On Schatten–von Neumann Classes and Trace Class Estimates

    • Fritz Gesztesy, Marcus Waurick
    Pages 23-33
  6. Pointwise Estimates for Integral Kernels

    • Fritz Gesztesy, Marcus Waurick
    Pages 35-53
  7. Dirac-Type Operators

    • Fritz Gesztesy, Marcus Waurick
    Pages 55-63
  8. Derivation of the Trace Formula: The Trace Class Result

    • Fritz Gesztesy, Marcus Waurick
    Pages 65-76
  9. Derivation of the Trace Formula: Diagonal Estimates

    • Fritz Gesztesy, Marcus Waurick
    Pages 77-99
  10. The Case n = 3

    • Fritz Gesztesy, Marcus Waurick
    Pages 101-105
  11. The Index Theorem and Some Consequences

    • Fritz Gesztesy, Marcus Waurick
    Pages 107-117
  12. Perturbation Theory for the Helmholtz Equation

    • Fritz Gesztesy, Marcus Waurick
    Pages 119-129
  13. The Proof of Theorem 10.2: The Smooth Case

    • Fritz Gesztesy, Marcus Waurick
    Pages 131-150
  14. The Proof of Theorem 10.2: The General Case

    • Fritz Gesztesy, Marcus Waurick
    Pages 151-156
  15. Back Matter

    Pages 167-194

About this book

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.

Authors and Affiliations

  • Dept of Mathematics, University of Missouri, Columbia, USA

    Fritz Gesztesy

  • Institut für Analysis, TU Dresden, Dresden, Germany

    Marcus Waurick

Bibliographic Information

Buy it now

Buying options

eBook USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book USD 49.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Other ways to access