  The Dirac Hamiltonian with the Aharonov-Bohm potential provides an example of
a non-Fredholm operator for which all spectral asymmetry comes entirely from
the continuous spectrum. In this case one finds that the use of standard
definitions of the resolvent regularized, the heat kernel regularized, and the
Witten indices misses the contribution coming from the continuous spectrum and
gives vanishing spectral asymmetry and axial anomaly. This behaviour in the
case of the continuous spectrum seems to be general and its origin is
discussed.
