  We develop by example a type of index theory for non-Fredholm operators. A
general framework using cyclic homology for this notion of index was introduced
in a separate article [CaKa13] where it may be seen to generalise earlier ideas
of Carey-Pincus and Gesztesy-Simon on this problem. Motivated by an example in
two dimensions in [BGG+87] we introduce in this paper a class of examples of
Dirac type operators on R^{2n} that provide non-trivial examples of our
homological approach. Our examples may be seen as extending old ideas about the
notion of anomaly introduced by physicists to handle topological terms in
quantum action principles with an important difference, namely we are dealing
with purely geometric data that can be seen to arise from the continuous
spectrum of our Dirac type operators.
