The Callias Index Formula Revisited (Lecture Notes in Mathematics Book 2157)

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.--Résumé de l'éditeur Front Matter....Pages i-ix Introduction....Pages 1-8 Notational Conventions....Pages 9-11 Functional Analytic Preliminaries....Pages 13-21 On Schatten–von Neumann Classes and Trace Class Estimates....Pages 23-33 Pointwise Estimates for Integral Kernels....Pages 35-53 Dirac-Type Operators....Pages 55-63 Derivation of the Trace Formula: The Trace Class Result....Pages 65-76 Derivation of the Trace Formula: Diagonal Estimates....Pages 77-99 The Case n = 3....Pages 101-105 The Index Theorem and Some Consequences....Pages 107-117 Perturbation Theory for the Helmholtz Equation....Pages 119-129 The Proof of Theorem 10.2: The Smooth Case....Pages 131-150 The Proof of Theorem 10.2: The General Case....Pages 151-156 A Particular Class of Non-Fredholm Operators L and Their Generalized Witten Index....Pages 157-165 Back Matter....Pages 167-194