  The square root of the positive definite hermitian operator $D_w^{\dagger}
D_w$ in Neuberger's proposal of exactly massless quarks on the lattice is
implemented by the recursion formula $Y_{k+1} = {1/2} (Y_k + D_w^{\dagger} D_w
Y_k^{-1})$ with $Y_0 = \Id$, where $Y_k^2$ converges to $D_w^{\dagger} D_w$
quadratically. The spectrum of the lattice Dirac operator for single massless
fermion in two dimensional background U(1) gauge fields is investigated. For
smooth background gauge fields with non-zero topological charge, the exact zero
modes with definite chirality are reproduced to a very high precision on a
finite lattice and the Index Theorem is satisfied exactly. The fermionic
determinants are also computed and they are in good agreement with the
continuum exact solution.
