Gesztesy F., Waurick M. The Callias Index Formula Revisited

Springer, 2016. — 191 p. — (Lecture Notes in Mathematics 2157). — ISBN 978-3-319-29976-1.Новый взгляд на формулу индекса КаллиасаThese 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.Notational Conventions
Functional Analytic Preliminaries
On Schatten–von Neumann Classes and Trace Class Estimates
Pointwise Estimates for Integral Kernels
Dirac-Type Operators
Derivation of the Trace Formula: The Trace Class Result
Derivation of the Trace Formula: Diagonal Estimates
The Case n = 3
The Index Theorem and Some Consequences11 Perturbation Theory for the Helmholtz Equation
The Proof of Theorem 10.2: The Smooth Case
The Proof of Theorem 10.2: The General Case
A Particular Class of Non-Fredholm Operators L and Their Generalized Witten Index
A Construction of the Euclidean Dirac Algebra