An index theorem of Callias type for pseudodifferential operators

TL;DR

This paper establishes an index theorem for families of pseudodifferential operators on manifolds with boundary, extending previous results by incorporating scattering calculus and topological K-theory techniques.

Contribution

It generalizes Callias-type index theorems to operators with asymptotically conic metrics, using Melrose's scattering calculus and K-theory methods.

Findings

Index determined by boundary symbolic data

Special case for families of Dirac operators

Extension of known index formulas to new operator classes

Abstract

We prove an index theorem for families of pseudodifferential operators generalizing those studied by C. Callias, N. Anghel and others. Specifically, we consider operators on a manifold with boundary equipped with an asymptotically conic (scattering) metric, which have the form D + i \Phi, where D is elliptic pseudodifferential with Hermitian symbols, and \Phi is a Hermitian bundle endomorphism which is invertible at the boundary and commutes with the symbol of D there. The index of such operators is completely determined by the symbolic data over the boundary. We use the scattering calculus of R. Melrose in order to prove our results using methods of topological K-theory, and we devote special attention to the case in which D is a family of Dirac operators, in which case our theorem specializes to give families versions of the previously known index formulas.