  We give an index formula for a class of Dirac operators coupled with
unbounded potentials. More precisely, we study operators of the form P := D+ V,
where D is a Dirac operators and V is an unbounded potential at infinity on a
possibly non-compact manifold M_0. We assume that M_0 is a Lie manifold with
compactification denoted M. Examples of Lie manifolds are provided by
asymptotically Euclidean or asymptotically hyperbolic spaces. The potential V
is required to be invertible outside a compact set K and V^{-1} extends to a
smooth function on M\K that vanishes on all faces of M in a controlled way.
Using tools from analysis on non-compact Riemannian manifolds, we show that the
computation of the index of P reduces to the computation of the index of an
elliptic pseudodifferential operator of order zero on M_0 that is a
multiplication operator at infinity. The index formula for P can then be
obtained from earlier results. The proof also yields similar index formulas for
Callias-type pseudodifferential operators coupled with bounded potentials that
are invertible at infinity on asymptotically commutative Lie manifolds, a class
of manifolds that includes the scattering and double-edge calculi.
