A General Index Theorem for Callias-anghel Operators

We prove a families version of the index theorem for operators generalizing those studied by C. Callias and later by N. Anghel, which are operators on a manifold with boundary having the form D + iΦ, where D is elliptic pseudodifferential with self-adjoint symbols, and Φ is a self-adjoint 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 Dirac operator, in which case our theorem specializes to reproduce the known index formulas, valid now for families of operators.