An index formula for perturbed Dirac operators on Lie manifolds

TL;DR

This paper derives an index formula for a class of Dirac operators with unbounded potentials on Lie manifolds, extending index theory to non-compact geometries with controlled behavior at infinity.

Contribution

It introduces a new index formula for perturbed Dirac operators on Lie manifolds with unbounded potentials, broadening the scope of index theory on non-compact spaces.

Findings

Index formula reduces to elliptic pseudodifferential operator on M_0

Applicable to asymptotically Euclidean and hyperbolic spaces

Extends to Callias-type operators with bounded potentials

Abstract

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.