  We study the role of the O(a)-improving clover term for the spectrum of the
lattice Dirac operator using cooled and thermalized SU(2) gauge field
configurations. For cooled configurations we observe improvement of the
spectral properties when adding the clover term. For the thermalized case
(12^4, beta = 2.4) without clover term we find a rather bad separation of
physical and doubler branches making a probabilistic interpretation of the
Atiyah-Singer index theorem on the lattice questionable for this beta and
lattice size. Adding the clover term leads to the creation of additional real
eigenvalues which come in pairs of opposite chirality thus further worsening
the situation for the index theorem.
