  Lower bounds on the magnitude of the spectrum of the Hermitian Wilson-Dirac
operator H(m) have previously been derived for 0<m<2 when the lattice gauge
field satisfies a certain smoothness condition. In this paper lower bounds are
derived for 2p-2<m<2p for general p=1,2,...,d where d is the spacetime
dimension. The bounds can alternatively be viewed as localisation bounds on the
real spectrum of the usual Wilson-Dirac operator. They are needed for the
rigorous evaluation of the classical continuum limit of the axial anomaly and
index of the overlap Dirac operator at general values of m, and provide
information on the topological phase structure of overlap fermions. They are
also useful for understanding the instanton size-dependence of the real
spectrum of the Wilson-Dirac operator in an instanton background.
