The Callias Index Formula Revisited

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Springer, Jun 28, 2016 - Mathematics - 192 pages
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.
 

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Contents

2 Notational Conventions
9
3 Functional Analytic Preliminaries
12
4 On Schattenvon Neumann Classes and Trace Class Estimates
23
5 Pointwise Estimates for Integral Kernels
34
6 DiracType Operators
55
The Trace Class Result
64
Diagonal Estimates
77
9 The Case n3
101
11 Perturbation Theory for the Helmholtz Equation
119
The Smooth Case
130
The General Case
151
14 A Particular Class of NonFredholm Operators L and Their Generalized Witten Index
157
A Construction of the Euclidean Dirac Algebra
167
B A Counterexample to Lemma 5Ca78
174
References
185
Index
190

10 The Index Theorem and Some Consequences
106

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