A Callias-type index theorem with degenerate potentials

TL;DR

This paper generalizes Callias' index theorem to self-adjoint Dirac operators with degenerate potentials on asymptotically conic manifolds, showing the index depends only on data at infinity despite potential nullspace degeneracies.

Contribution

It extends the classical Callias index theorem to include potentials with constant rank nullspaces at infinity, broadening its applicability.

Findings

Derived a new index formula for degenerate potentials

Showed the index depends only on asymptotic data

Extended the theorem to a broader class of manifolds

Abstract

A generalization of Callias' index theorem for self adjoint Dirac operators with skew adjoint potentials on asymptotically conic manifolds is presented in which the potential term may have constant rank nullspace at infinity. The index obtained depends on the choice of a family of Fredholm extensions, though as in the classical version it depends only on the data at infinity.