  We investigate the index of the Neuberger's Dirac operator in abelian gauge
theories on finite lattices by numerically analyzing the spectrum of the
hermitian Wilson-Dirac operator for a continuous family of gauge fields
connecting different topological sectors. By clarifying the characteristic
structure of the spectrum leading to the index theorem we show that the index
coincides to the topological charge for a wide class of gauge field
configurations. We also argue that the index can be found exactly for some
special but nontrivial configurations in two dimensions by directly analyzing
the spectrum.
