  We study perturbed Dirac operators of the form $ D_s= D + s\A
:\Gamma(E^0)\rightarrow \Gamma(E^1)$ over a compact Riemannian manifold $(X,
g)$ with symbol $c$ and special bundle maps $\A : E^0\rightarrow E^1$ for
$s>>0$. Under a simple algebraic criterion on the pair $(c, \A)$, solutions of
$D_s\psi=0$ concentrate as $s\to\infty$ around the singular set $Z_\A$ of $\A$.
We prove a spectral separation property of the deformed Laplacians $D_s^*D_s$
and $D_s D_s^*$, for $s>>0$. As a corollary we prove an index localization
theorem.
