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We give an index formula for a class of Dirac operators coupled withunbounded 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 apossibly non-compact manifold M_0. We assume that M_0 is a Lie manifold withcompactification denoted M. Examples of Lie manifolds are provided byasymptotically Euclidean or asymptotically hyperbolic spaces. The potential Vis required to be invertible outside a compact set K and V^{-1} extends to asmooth 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 thecomputation of the index of P reduces to the computation of the index of anelliptic pseudodifferential operator of order zero on M_0 that is amultiplication operator at infinity. The index formula for P can then beobtained from earlier results. The proof also yields similar index formulas forCallias-type pseudodifferential operators coupled with bounded potentials thatare invertible at infinity on asymptotically commutative Lie manifolds, a classof manifolds that includes the scattering and double-edge calculi.

An index formula for perturbed Dirac operators on Lie manifolds