An Index Formula For Perturbed Dirac Operators On Lie Manifolds

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 noncompact manifold M0. We assume that M0 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 such that V is invertible outside a compact set K and V .1 extends to a smooth function on M rK 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 M0 that is a multiplication operator at infinity. The index formula for P can then be obtained from the results of [17]. The proof also yields similar index formulas for Dirac 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.