An Index Formula for Perturbed Dirac Operators on Lie Manifolds

We prove an index formula for a class of Dirac operators coupled with unbounded potentials, also called "Callias-type operators". More precisely, we study operators of the form P := D + V, where D is a Dirac operator and V is an unbounded potential at infinity on a non-compact manifold M-0. We assume that M-0 is a Lie manifold with compactification denoted by M. Examples of Lie manifolds are provided by asymptotically Euclidean or asymptotically hyperbolic spaces and many others. The potential V is required to be such that V is invertible outside a compact set K and V-1 extends to a smooth vector bundle endomorphism over 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 the results of Carvalho (in K-theory 36(1-2): 1-31, 2005). As a first step in the proof, we obtain a similar index formula for general pseudodifferential operators coupled with bounded potentials that are invertible at infinity on a restricted class of Lie manifolds, so-called asymptotically commutative, which includes, for instance, the scattering and double-edge calculi. Our results extend many earlier, particular results on Callias-type operators.