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.
